## 1. Introduction

Rolling element bearing is a key component of rotating machines. In many cases, it operates with high speed, heavy load, and prolonged time. Thus, such localized defects as pitting, spalling, fretting, scuffing may generate on the contact surfaces. These faults will lead to abnormal vibrations, which will reduce the working accuracy and even cause catastrophic accidents [

1,

2]. Accordingly, the detection of the bearing failures as early as possible is an important and meaningful subject for mechanical maintenance. Vibration-based diagnostic of rolling element bearings has been interestingly investigated in recent decades [

3]. As the bearing rotates, the fault-induced impulses will repetitively appear with a specific frequency that is decided by location of the defect. Generally, they are of short duration and their responses are hardly directly identified in the temporal waveform or frequency spectrum due to the background noises from machine [

4]. Fortunately, these repetitive transients are excited through a resonance of the system at a much higher frequency band, which amplifies and preserves the fault information [

5]. It implies that band-pass filtering the vibration signal around the resonant frequency can extract the fault features as well as eliminate the interferences. This approach is generally called the high-frequency resonance technique or envelope analysis, as illustrated in

Figure 1. Hence, the most critical challenge in envelope analysis can be drawn as the informative frequency band (IFB) selection [

6], which has attracted a large amount of attention in recent years [

7,

8,

9,

10,

11].

Spectral kurtosis (SK) [

12], as well as its fast implementation called as kurtogram [

13], were viewed as a milestone for the IFB selection, in which the kurtosises of the coefficients at the output of quasi-analytic filter-banks with different central frequencies and bandwidths were employed in a frequency/frequency resolution plane for representing and detecting non-stationarities in the vibration signal. Once proposed, it has been paid a lot of attention and effort for its improvement for fault detection of rotating machines. In [

14], an improved kurtogram was proposed by adopting wavelet packet transform (WPT) to replace the FIR filters to improve the accuracy of identifying the IFB and the signal-to noise ratio (SNR) of the filtered signal. After that, a new statistical index that is based on alpha-stable distribution was applied in [

15] to substitute for kurtosis to characterize the non-Gaussian of fault signals. In [

16], a sparsogram was further proposed while using the sparsity measurements to select the wavelet packet node from WPT. Moreover, another alternative estimator, called the Gini index, was introduced in [

17] to improve the resistance of kurtosis-guided-grams to random-impulse. Recently, spectral L2/L1 norm was given in [

18] to explain the SK and spectral correlation in a comprehensive way and in [

19] the author compared it with kurtosis, smoothness index, and Gini index. The conclusion is that all of these popular sparsity indexes may be unfortunately affected by outliers; only some are less sensitive to outliers than others. In these above improvements of SK, the indexes are all utilized to quantitatively evaluate the band-pass filtering signal or its analytic envelope to detect the impulses can be categorized as time-domain estimators, whose biggest disadvantage is vulnerability to the outliers, as mentioned above.

Correspondingly, some frequency-domain approaches have also been proposed from another perspective to improve SK. A prime example is the protrugram [

20], which selects the IFB based on the kurtosis of the envelope spectrum other than kurtosis of the time-domain demodulated signal to eliminate the negative effect of non-Gaussian outliers in the fault signals. As far as the authors know, it firstly applies the frequency-domain sparsity to characterize the fault impulses. Approximatively, an enhanced SK was put forward in [

21], where the kurtosis of the power spectrum of the envelope of the filtered signal by WPT at each node was calculated to construct the kurtogram. However, the frequency-domain sparsity might result from periodic interferences, such as rotor eccentricity or gear meshing and results in fewer harmonics of fault characteristic frequencies (FCFs) in the envelope spectrum. These two aspects are all unwanted for bearing fault diagnosis. After that, measuring cyclostationarity [

22] rather than the impulsiveness in IFB selection began to get more attention. In [

23], kurtosis of the autocorrelation of the squared envelope (SE) rather than the direct filtered signal was utilized to generate the autogram for bearing fault diagnosis. In [

24], a series of cyclostationary indicators that were based on the generalized likelihood ratio were proposed under the hypothesis of generalized Gaussian signals. In a recent work [

25], a new IFB selection method that was based on the log-envelope spectrum, called log-cycligram, was given to capture the cyclostationarity separately from non-Gaussianity. Objectively, the effectiveness of the newly proposed methods [

24,

25] depends on the FCFs as prior knowledges. Little difference between the calculated FCF and the actual in the spectrum was required for extracting the specific repetitive transients to some extent. This flaw also existed in our previous study [

26], in which the correlated kurtosis (CK) of the power spectrum was used as a novel estimator to replace kurtosis. From the above review, it can be concluded that a single time-domain, frequency-domain, or cyclostationary indicator is incomplete for bearing fault features extraction.

In recently, another interesting method that was motivated from the concepts of thermodynamics, named infogram, was proposed in [

27]. As the author introduced, the negentropy of the SE of the filtered signal was employed to characterize the time-domain impulsiveness by SE infogram and negentropy of the squared envelope spectrum (SES) was employed to characterize the frequency-domain one by SES infogram. Furthermore, the average of SE infogram and SES infogram was employed to generate the average (AVE) infogram to characterize both the impulsive and cyclostationary signatures of the repetitive transients, according to Hirschman’s uncertainty principle.

The aforementioned study gave a combination of time and frequency-domain estimators for IFB selection. However, there are also limitations that are associated with it. On one hand, infogram divide the spectrum using fixed boundaries of the filter-bank like kurtogram, it might occasionally mismatch the IFB [

28]. In [

9], accugram was proposed based on infogram in which the health data was utilized as reference to improve its accuracy. Another effective way to deal with that is to extend the fast filtering method to optimal filtering based on evolutionary optimization [

29,

30,

31]. In [

32], the infogram was extended to Bayesian inference based optimal wavelet filter for identifying the localized defects. However, only negentropy of SE was adopted in the measurement function. It againsts the initial desire of infogram to both take impulsiveness and cyclostationarity into consideration. On the other hand, the AVE infogram was obtained by the mean of an SE infogram and SES infogram. It does not fundamentally overcome the drawback that the time-domain negentropy is not immune to the impulsive noise and the frequency-domain one is not immune to the cyclostationary noise [

33,

34]. However, IFB selection is easily affected by impulsive or cyclostationary noise. In addition, the impulsiveness and the cyclostationarity of repetitive transients are competitive. Hence, the sparsity of the time-domain and frequency-domain should achieve a relative balance i.e., a compromise in IFB selection. Therefore, how to balance the impulsiveness and cyclostationarity of the repetitive transients but avoid the complex interferences effectively are still worth further consideration.

A negentropy-induced multi-objective optimized wavelet filter is proposed for extracting the repetitive transients to solve the aforementioned problems. The parameters of an anti-symmetric real Laplace wavelet (ARLW) are optimized with two objective functions: maximum of the time and frequency-domain negentropy in order to characterize the impulsive and cyclostationary features, respectively. With the help of competition mechanism of multi-objective grey wolf optimizer (MOGWO) [

35], some Pareto optimal solutions will be generated that are non-dominated by the others. It implies that the solution representing the most impulsive or cyclostationary components has been eliminated at iterations. Subsequently, the IFB can be easily identified by the maximum average negentropy of the Pareto set. The main contributions of the proposed method can be concluded, as follows: (1) The IFB selection is directly modeled as a multi-objective optimization problem, and it is the first attempt to solve it in bearing fault diagnosis while using the MOGWO. (2) The average negentropies of the non-dominated solutions are utilized to extract and balance the fault features. Furthermore, the maximum of average negentropies can be directly utilized to find the single optimal solution from the Pareto ones for IFB selection. It is simple enough than the knee point selection methods.

The rest of the paper is organized, as follows. In

Section 2, the detail of the proposed methodology that is based on wavelet filtering and MOGWO algorithm is introduced. Next, its performances are examined and comparatively studied using several experimental signals, including two cases of incipient faults diagnosis and a case of IFB tracking in

Section 3. Finally,

Section 4 summarizes some conclusions.