IGIgram: An Improved Gini Index-Based Envelope Analysis for Rolling Bearing Fault Diagnosis

: The transient impulse features caused by rolling bearing faults are often present in the resonance frequency band which is closely related to the dynamic characteristics of the machine structure. Informative frequency band identi ﬁ cation is a crucial prerequisite for envelope analysis and thereby accurate fault diagnosis of rolling bearings. In this paper, based on the ratio of quasi-arithmetic means and Gini index, improved Gini indices (IGIs) are proposed to quantify the transient impulse features of a signal, and their effectiveness and advantages in sparse quanti ﬁ cation are con ﬁ rmed by simulation analysis and comparisons with traditional sparsity measures. Furthermore, an IGI-based envelope analysis method named IGIgram is developed for fault diagnosis of rolling bearings. In the new method, an IGI-based indicator is constructed to evaluate the impulsiveness and cyclostationarity of the narrow-band ﬁ ltered signal simultaneously, and then a frequency band with abundant fault information is adaptively determined for extracting bearing fault features. The performance of the IGIgram method is veri ﬁ ed on the simulation signal and railway bearing experimental signals and compared with typical sparsity measures-based envelope analysis methods and log-cycligram. The results demonstrate that the proposed IGIs are ef ﬁ cient in quantifying bearing fault-induced transient features and the IGIgram method with appropriate power exponent can effectively achieve the diagnostics of different axle-box bearing faults.


I. INTRODUCTION
Rolling bearings are widely used in rotating machinery equipment in the fields of transportation, energy and chemical industries, such as high-speed trains, aero engines, automobile engines, wind turbine generator sets, and steam turbine generator sets. Unexpected damage or failure in rolling bearings may lead to serious human injury or huge economic losses [1]. Thus, fault diagnosis of rolling bearings in key transportation, energy and chemical equipment is of great significance to ensure operational safety and improve economic benefits. Up to now, many signal processing-based methods have been developed for fault diagnosis of rolling bearings, such as blind deconvolution [2], signal decomposition [3], envelope analysis [4], stochastic resonance [5,6], morphological filtering [7], cyclic spectral analysis [8,9] and modulation signal bispectrum [10]. Among these methods, envelope analysis is one of the most commonly used methods for fault diagnosis of rolling bearings and other rotating mechanical components. In this method, band-pass filtering is usually performed on a resonance frequency band to extract the fault-caused transient impulses hidden in the noisy vibration signals [11]. Therefore, informative frequency band identification (IFBI) is a crucial prerequisite for rolling bearing fault diagnosis based on envelope analysis.
At present, many IFBI methods have been established for fault diagnosis of rolling bearings, which can be roughly divided into blind methods and targeted methods [12]. It should be emphasized that the "blind" and "targeted" here are only for the characteristics of the frequency band identification criteria rather than the method itself and its implementation process. This is a key point to note so as not to cause ambiguity.
The blind IFBI method makes no assumptions about the periodicity of the bearing fault features in the analyzed signal [12], and can adaptively identify an informative frequency band (IFB) closely related to transient fault features. A representative blind IFBI method is the fast kurtogram proposed by Antoni [4]. In this method, a 1/3binary tree filter bank is designed to split the vibration signal into a group of narrowband filtered signals with different center frequencies and bandwidths, and the kurtosis of the narrowband filtered signal is taken as a criterion to identify the IFB by evaluating the impulsiveness of bearing fault features. However, the fast kurtogram tends to select the fault-unrelated frequency band when confronted with vibration signals polluted by strong random impulse noise, which may attribute to the inherent limitation of kurtosis susceptible to outliers. Subsequently, the protrugram [13] was developed for IFBI by employing the kurtosis of the envelope spectrum of the narrowband filtered signal as the evaluation criterion of the fault information. The autogram was proposed for bearing diagnosis in [14] by using the kurtosis of the autocorrelation of the squared envelope of the filtered signal as the frequency band identification criterion. Similarly, correlation spectral negentropy was proposed in [15] to identify IFB. Since the protrugram is dedicated to evaluating the cyclostationarity rather than the impulsiveness of bearing fault features and the autogram aims to characterize the periodicity of bearing fault features, they are more robust to random impulse noise in comparison with the fast kurtogram. However, the protrugram delivers low robustness to discrete harmonic interference from shaft rotation or gear meshing, which exhibits a high-level peak in the envelope spectrum; the autogram has difficulty extracting the desired repetitive transient fault features when strong discrete harmonics are present. Tse and Wang [16] used the ratio of L2 to L1 norm (L2/L1) of the power spectrum of the narrowband filtered signal to determine IFB for bearing fault diagnosis. However, L2/L1 is also susceptible to large outliers, thus its normalized version namely Hoyer index (HI) [17] may be an alternative evaluation criterion. Antoni [18] fused both the impulsiveness and cyclostationarity into the infogram using the arithmetic mean of the negentropy (NE) of the squared envelope and the NE of the squared envelope spectrum (SES) for IFBI. Hebda-Sobkowicz et al. [19] replaced the arithmetic mean of partial infograms with the logarithmic mean, geometric mean and normalizing of partial infograms to improve the robustness of the classical infogram to non-Gaussian noise. HI and NE have weaker sensitivity to random impulses compared with kurtosis, but they also show poor IFBI performance in the presence of strong random impulses. In addition, Gini index (GI), originally used in the field of economics, was introduced into the field of mechanical fault diagnosis to quantify impulse features and guide the selection of IFB [20,21]. GI is less sensitive to non-Gaussian noise than kurtosis, NE and HI, whereas its fault-impulse discernibility is weaker than kurtosis, NE and HI [22]. Thus, it is necessary to develop robust transient feature quantification metrics for improving the accuracy of frequency band identification. In addition, the L-kurtosis [23], stability index [24], conditional variance statistic [25], averaged local kurtosis [26] and subband averaging kurtogram [27] were proposed to distinguish bearing fault-related frequency band in the presence of non-Gaussian noise. The works in [21,22,28,29] summarized and compared the performance of typical blind IFBI methods in bearing fault diagnosis.
The targeted IFBI method focuses on the (quasi) periodicity of bearing fault features to extract the cyclic components in the signal to be analyzed, requiring knowledge of the impulse period (in the time domain) or the fault characteristic frequency (in the frequency domain) before identifying IFB. For example, the harmonic-to-noise ratio [30] and correlated kurtosis [31,32] using the impulse period have been employed to construct the IFBI methods for extracting periodic bearing fault features. In recent years, the cyclostationarity of bearing fault signals has received increasing attention, and the IFBI methods using fault characteristic frequency have been continuously proposed, such as the ratio of cyclic content [33], indicator of second-order cyclostationarity [34], frequency domain correlated kurtosis [35] and weighted cyclic harmonic-to-noise ratio [36]. These methods mainly use the SES and the fault characteristic frequency of interest to construct the evaluation indicator of cyclostationarity as the frequency band identification criterion. Inspired by the works in [37,38] where the log-envelope spectrum was discovered to be a robust tool for characterizing cyclostationarity in the presence of high impulse noise, the IFBI methods based on the log-envelope spectrum were established, such as log-cycligram (LC) [12], cyclic harmonic ratio [39] and LEASgram [40]. In addition, by exploiting the insensitivity of correntropy to impulse noise, FECgram [41] was proposed to identify IFB for bearing diagnostics. The works in [12,42] comparatively investigated the performance of the blind and targeted IFBI methods in distinguishing IFB. Benefiting from the frequency band identification criteria using impulse period or fault characteristic frequency, the targeted IFBI methods are generally more robust than the blind IFBI methods in the presence of high impulsive noise. However, the targeted IFBI methods require an accurate impulse period or fault characteristic frequency as a prerequisite to ensure the accurate identification of fault-related frequency bands. In contrast, the blind IFBI methods do not require impulse period and fault characteristic frequency in distinguishing fault-related frequency band, thus exhibiting wider adaptability. Therefore, this paper aims to develop a blind IFBI method which is robust to impulse noise and can achieve fault diagnosis performance similar to those of the targeted IFBI methods.
In recent years, the design of new quantitative indicators of machine fault features has received increasing attention. Wang et al. [43] proposed the sum of weighted normalized squared envelope as a framework to characterize the repetitive transients for machine condition monitoring. Hou et al. [44] discovered that typical sparsity measures can be reformulated into the ratio of different quasi-arithmetic means (RQAM). RQAM has been proved to be a powerful tool for designing quantitative indicators for characterizing repetitive transients [44][45][46]. To improve the accuracy of IFBI for bearing fault diagnosis, the design of new sparsity measures and their application in bearing fault diagnosis are investigated. The main novelties and contributions of this paper include: (1) Based on RQAM, a novel generalization method of GI is developed, which allows the single parameter generalization of classic GI by using the power function-based quasi-arithmetic means and leads to new quantitative indicators named improved GIs (IGIs) for quantifying transient features.
(2) The sparse quantization capability and repetitive transient discriminability of IGIs are investigated using simulations, and the results show that IGIs exhibit rich sparse quantization capabilities and can achieve improved repetitive transient discriminability compared with traditional sparsity measures.
(3) An IGI-based IFBI method named IGIgram is proposed for bearing fault diagnosis, which utilizes IGI to adaptively identify an IFB with rich fault information by evaluating both the impulsiveness and cyclostationarity of bearing fault features.
(4) The effectiveness of IGIgram is validated using simulation and experimental signals, and the results confirm that IGIgram with appropriate power exponent can achieve accurate IFBI for effective extraction of repetitive transients and fault diagnosis of railway axle-box bearings.
The remainder of this paper is arranged as follows. In Section II, the traditional sparsity measures for IFBI are briefly reviewed and the formal definition and performance analysis of IGIs are presented. Section III introduces the envelope analysis method based on IGIs for bearing fault diagnosis. In Sections IV and V, the simulation signals and railway axlebox bearing experimental signals are analyzed to validate the effectiveness of the proposed fault diagnosis method, and the performance is compared with the IFBI methods based on typical sparsity measures. Section VI presents the comparison results with the LC method. The main conclusions of this paper are summarized in Section VII.

II. TRADITIONAL AND NOVEL SPARSITY MEASURES
A. TRADITIONAL SPARSITY MEASURES Sparsity measures are usually regarded as the measure of the energy concentration of a signal [47]. Specifically, if all energy is concentrated in one element of the signal, the sparsity value of the signal is maximum. On the contrary, if all elements have the same energy, the sparsity of the signal is minimum. Let x = ½x 1 ,x 2 , · · · ,x N be a discrete vibration signal with N samples, the typical sparsity measures for characterizing transient impulse features in IFBI are introduced as follows. Kurtosis (Kurt) [48]: Negentropy [18]: Ratio of L2 to L1 norm (L2/L1) [16]: Hoyer index [17]: Gini index [49]: where kxk 1 and kxk 2 denote the L1 norm and L2 norm, respectively; x asc = ½x ð1Þ ,x ð2Þ , · · · ,x ðNÞ is the permutation of sequence x = ½x 1 ,x 2 , · · · ,x N in ascending order, i.e., x ð1Þ ≤ x ð2Þ ≤ · · · ≤ x ðNÞ . Note that the input sequence of GI is nonnegative. These sparsity measures are typical indicators of transient impulse features in a signal and have been applied to the condition monitoring and fault diagnosis of rolling bearings by characterizing the impulsiveness or cyclostationarity of bearing fault signals. However, traditional sparsity measures have exposed some shortcomings in practical applications. The value of kurtosis varies largely with the signal amplitude, especially outliers, which is not conducive to the detection of bearing fault impulses in industrial scenarios. Moreover, the range of kurtosis cannot be accurately estimated. The sensitivity of NE, L2/L1 and HI to outliers is lower than kurtosis, but their fault impulse detection performance will still be affected when encountering strong random impulses [22,50]. GI is robust to outliers and has a clear range of [0,1], but its ability to identify bearing fault impulses under noise interference is weaker than kurtosis and NE [22]. Therefore, it is necessary to develop new sparsity measures that are robust to random impulses and background noise for transient feature characterization and bearing fault diagnosis.

B. IMPROVED GINI INDICES
To overcome the limitations of traditional sparsity measures in characterizing transient impulse features and detecting bearing faults, new sparsity measures or statistical indicators have been explored recently. Based on RQAM, Hou et al. [45] proposed generalized Gini indices (GGIs) by introducing the nonlinear weights into the classic GI, as follows: where w n = ½2ðN − nÞ þ 1 a = P N n=1 ½2ðN − nÞ þ 1 a is the nonlinear weight and a > 0 is the weight parameter.
Compared with GI and other traditional sparsity measures, GGIs can achieve different sparse quantization capabilities and tunable random transient resistibility by adjusting weight parameter. However, the repetitive transient discriminability of GGIs with different weight parameters shows small differences and is inferior to Kurt, NE and HI, which will be elucidated in Section II.C. Moreover, both GI and GGIs employ a line function as the generator of the quasi-arithmetic means. Given the nonlinear property of power functions, it is worth investigating the construction of sparsity measures by using quasi-arithmetic means based on power function. Therefore, based on RQAM and classical GI, IGIs are proposed as a new family of statistical indicators, as follows: where kxk p is the Lp norm, p > 0 is the power exponent, w 0 n = ½2ðN − nÞ þ 1=N 2 and P N n=1 w 0 n = 1. IGIs are reduced to GI when the power exponent p = 1. The power exponent of IGIs allows their performance or capability to be tunable. In addition, the definition in Eq. (7) indicates the following important properties: • IGIs are scale-invariant, which means that the amplitude of each element multiplied by a constant factor will not change IGIs of a signal.
• IGIs are positive and their values are always between 0 and 1.
A numerical example is used to illustrate the motivation for using power functions to construct quasi-arithmetic means, as shown in Fig. 1. Fig. 1a displays a non-negative sequence (p = 1) with the largest element greater than 1 and its square (p = 2) and square root (p = 0.5). Fig. 1b displays a non-negative sequence with the largest element less than 1 and its square (p = 2) and square root (p = 0.5). Two prominent impulses can be observed in the original sequence (i.e., power exponent p = 1). For sequences with power exponents p = 1, 2 and 0.5, the ratios of the largest element to the smallest element of the three sequences are 72.88, 5310.97 and 8.54, respectively. Note that the quantization results for the sequences shown in Fig. 1a, b are the same. It can be discovered that the nonlinear characteristic of the power function can be used to non-uniformly scale the sequence amplitude. Furthermore, increasing the power exponent p of the sequence can increase the salience of the impulses relative to other elements, regardless of whether the largest element of the non-negative sequence is greater than 1.
As shown in Eq. (7), a linear weight sequence is applied to the normalized sequence x p ð1Þ =kxk p p , x p ð2Þ = kxk p p , : : : ,x p ðNÞ =kxk p p when calculating the IGI of the sequence. Since increasing the power exponent p can increase the salience of the impulses relative to other elements, it can increase the sensitivity of the IGI to transient features. Conversely, decreasing the power exponent p can reduce the salience of the impulses relative to other elements, thereby increasing the resistibility of the IGI to random transients. Therefore, the power exponent of IGI allows its performance or capability to be tunable. The following numerical analysis will confirm these properties of IGI.

C. PERFORMANCE ANALYSIS
This section illustrates the important properties of IGIs in transient feature characterization by simulation signals, and the performance is compared with existing sparsity measures, including Kurt, NE, L2/L1, HI, GI and GGIs. Bernoulli distributions with different success probabilities are often used to generate data sequences with different sparsity [45,47]. To evaluate the sparse quantification capability of IGIs, a series of data sequences with different sparsity were generated when the success probability of the Bernoulli distribution is increased from 0.005 to 1 in increments of 0.005. The length of each data sequence is 20,000 data points. Figs. 2 and 3 show the GGIs with different weight parameters and the IGIs with different power exponents of the data sequences generated by the Bernoulli distribution with different success probabilities, respectively. The weight parameter a of GGIs includes increasing from 0.1 to 1 in increments of 0.1 and from 1.5 to 10 in increments of 0.5. The power exponent p of IGIs includes increasing from 0.1 to 1 in increments of 0.1 and from 1.5 to 10 in increments of 0.5. Five traditional sparsity measures are also displayed for comparison. Note that all indicators are scaled between 0 and 1.
When the success probability of the Bernoulli distribution increases from 0.005 to 1, the sparsity of the data sequence gradually becomes weaker. Similar to GGIs, IGIs with different power exponents can monotonically quantify the sparsity of a data sequence, i.e., increase gradually as the sparsity of the data sequence increases. It can be observed that the sparse quantization capability of IGIs can be tuned by changing the power exponent. In addition, the descending gradient of the sparse quantization curve can reflect the random transient resistibility of the indicator. The smaller the descending gradient, the stronger the random transient resistibility of the indicator. The results show that decreasing the power exponent can improve the random transient resistibility of IGIs, and IGIs with appropriate power exponents show stronger random transient resistibility than the five traditional sparsity measures. Therefore, IGIs and GGIs exhibit similar characteristics in random transient resistibility.
Repetitive transient features caused by defects in rotating machinery are often contaminated by background noise,   show that the repetitive transient discriminability of GGIs with different weight parameters is similar to that of classical GI and inferior to Kurt, NE and HI. Reducing the weight parameter can slightly improve the repetitive transient discriminability of GGIs, but the effect of the weight parameter on improving the repetitive transient discriminability of GGIs is limited. In contrast, IGIs with different power exponents exhibit rich repetitive transient discriminability and can achieve stronger repetitive transient discriminability than five traditional sparsity measures and GGIs with different weight parameters, as exhibited in Figs. 5 and 6b. In addition, increasing the power exponent can effectively improve the repetitive transient discriminability of IGIs, but too high power exponents make IGIs susceptible to interference noise.
The above results lead to the following important remarks: (1) IGIs with different power exponents can quantify the sparsity of data sequences; (2) IGIs with appropriate power exponents can simultaneously possess strong random transient resistibility and repetitive transient discriminability; (3) IGIs with appropriate power exponents can achieve excellent transient feature quantization compared to traditional sparsity measures.

III. IMPROVED GINI INDEX-GUIDED ENVELOPE ANALYSIS
Recently, Hebda-Sobkowicz et al. [19] investigated the extension of the classic infogram by substituting the   IGIgram: An Improved Gini Index-Based Envelope Analysis arithmetic mean with the weighted mean, logarithmic mean and geometric mean for IFBI. The experimental results showed that the infogram based on the geometric mean (called geometric infogram) performs better than the arithmetic mean and logarithmic mean in bearing diagnostics. Inspired by the geometric infogram, the geometric mean of squared envelope IGI and squared envelope spectrum IGI (abbreviated as GMIGI) is proposed as the new criterion for IFBI, as follows: where IGI SE and IGI SES are the IGI values calculated from the squared envelope and squared envelope spectrum (SES), respectively. In envelope analysis, frequency band division is the prerequisite for IFBI to extract fault impulse features from noisy machine vibration signals. The 1/3-binary tree filter bank [4] has the merits of high computational efficiency and variable bandwidth, thus it is adopted in this study for frequency band division. For each decomposition level k of the filter bank, k = 0,1,1.6,2,2.6,3, · · · , the full frequency band ½0,F s =2 is split into 2 k narrow frequency bands with equal bandwidth F s =2 k+1 , where F s denotes the sampling frequency, as shown in Fig. 7. Further, a total of 2 k narrowband filtered signals can be obtained correspondingly by band-pass filtering. The details of the 1/3-binary tree filter bank are reported in [4].
On the basis of frequency band division, GMIGI is employed to evaluate the bearing fault features contained in the narrowband filtered signals. Finally, the GMIGI values of different narrow bands can form a two-dimensional map, named IGIgram, which can be used to identify an optimal narrowband filtered signal with abundant transient impulse features for fault diagnosis. A flowchart of IGI-guided envelope analysis for bearing fault diagnosis is shown in Fig. 8. The implementation procedure of the proposed fault diagnosis method is described as follows: Step 1: Collect the bearing vibration signals using data acquisition equipment and acceleration sensors.
Step 2: Conduct band-pass filtering on the measured vibration signal by the 1/3-binary tree filter bank to generate a group of narrowband filtered signals with different bandwidths and center frequencies.
Step 3: Calculate the squared envelope and SES of the narrowband filtered signal and calculate the GMIGI value of each narrowband filtered signal.
Step 4: Select the optimal narrowband filtered signal from the obtained candidates by the maximum GMIGI criterion.
Step 5: Perform SES analysis on the selected narrowband filtered signal and identify bearing faults based on characteristic fault frequencies.

IV. SIMULATION VERIFICATION A. NUMERICAL MODEL OF BEARING FAULT SIGNAL
A numerical simulation signal of bearing inner race fault is generated to validate the effectiveness of the IGIgram method in extracting repetitive transient features. The simulation signal considers the repetitive transient impulses caused by bearing defect shown in Eq. (13) and interference noises often encountered in industrial scenarios, including random transients in Eq. (14), discrete harmonics in Eq. (15) and background noise, as follows: xðtÞ = x 1 ðtÞ + x 2 ðtÞ + x 3 ðtÞ + nðtÞ (12) x 3 ðtÞ = 0.1½sinð2π · 10t þ π=6Þ þ sinð2π · 20t − π=3Þ (15)  A j = 2a j , a j ∼Nð1,0.1Þ (17) τ j ∼Uð0,1Þ (18) where Δ i is the time jitter caused by the slippage effect of the rolling element [10,51]; A j and τ j denote the amplitude and occurrence time of the jth random transient, respectively; Nð· , ·Þ and Uð· , ·Þ represent a normal distribution and a uniform distribution, respectively. In addition, the Gaussian noise with an SNR of −12.5 dB is added to simulate the background noise. The fault characteristic frequency of the bearing inner race is assumed to be 97 Hz. The sampling frequency and length are set to 12,800 Hz and 1 s, respectively.

B. SIMULATION RESULTS
The simulation signal of bearing inner race fault and its magnitude spectrum and SES are depicted in Fig. 9. Only two discrete harmonics, 10 Hz and 20 Hz, can be detected in the magnitude spectrum, and the inner race fault characteristic frequency f i = 97 Hz and its harmonics cannot be observed in SES.
To extract the repetitive transients, IGIs with power exponents p = 0.1, 0.3, 0.5, 0.8, 2, 3, 4 and 6 are introduced into the IGIgram method to select IFB. For comparison, Kurt, L2/L1, HI, GI and GGIs with a = 0.1, 0.3, 0.5, 0.8, 2, 3, 4 and 6 are employed to replace NE in the geometric infogram for selecting IFB. The decomposition level of the entire frequency band is set to 4 for all methods. Table I shows the frequency bands selected by different criteria on the simulation signal of bearing inner race fault. It can be observed that four different frequency bands are selected by these indicators. As some examples, Fig. 10 displays the geometric kurtogram, geometric infogram, IGIgram with p = 0.8 and IGIgram with p = 3. Furthermore, the envelopes and SESs of the band-pass filtered signal obtained from these four frequency bands are presented in Figs. 11 and 12, respectively.
Two large random impulses are exhibited in Fig. 11a, showing that Kurt and L2/L1 select the resonance frequency band of random impulses and fail to identify the resonance frequency band of repetitive transients.   In contrast, the repetitive transient features can be observed in Fig. 11b, d and the inner race fault characteristic frequency f i = 97 Hz and its harmonics can be easily detected in Fig. 12b, d. These results indicate that NE, HI, GI, GGIs with a = 0.1, 0.3, 0.5 and 0.8 and IGIs with p = 2, 3, 4 and 6 discriminate the correct resonance frequency bands (5,000, 400) and (5,067, 533). In addition, due to the wider frequency band, IGIs with p = 3, 4 and 6 achieve relatively good feature extraction results. The above results preliminarily verify the effectiveness of IGIs in identifying IFBs and extracting repetitive transient features.

V. EXPERIMENTAL RESULTS
In this section, the vibration signals acquired from railway axle-box bearings under different operating conditions are used to validate the efficiency of the proposed fault diagnosis method, and the performance is compared with the improved geometric infogram based on state-of-the-art sparsity measures. The experimental signals were collected from a railway axle-box bearing test rig. The test rig mainly consists of the driving device, wheelset mounted with axle-box bearings and loading device, as shown in Fig. 13a. The outer race fault and rolling element fault bearings were tested under different operating conditions. All bearing faults were implanted artificially with a depth of 0.2 mm and a width of 0.6 mm, as exhibited in Fig. 13b, c. The pitch diameter, rolling element diameter, number of rolling elements and contact angle of the tested axle-box bearings were 187.2 mm, 26.7 mm, 17 and 12.08°, respectively. Accelerometers were mounted along the vertical direction of the axle box to collect the vibration signals. The sampling

A. BEARING OUTER RACE FAULT
In this case, the vibration acceleration signal of the railway axle-box bearing with outer race fault is analyzed. The vibration signal was collected at a constant speed of 590 rpm, as shown in Fig. 14a. The transient impulse features induced by the bearing outer race fault are completely buried in the strong background noise, making it difficult to detect the fault characteristic frequency of the bearing outer race (f o = 71.88 Hz) and its harmonics from SES shown in Fig. 14c.
The IGIgram using IGIs with p = 0.1, 0.3, 0.5, 0.8, 2, 3, 4 and 6 are employed to extract the fault impulse features of bearing outer race. The geometric infogram and its improved methods using Kurt, L2/L1, HI, GI and GGIs with a = 0.1, 0.3, 0.5, 0.8, 2, 3, 4 and 6 are also applied to the signal in Fig. 14a for comparison. The decomposition level of the entire frequency band is set to 4 for all methods. Table II shows the six frequency bands selected by different criteria for bearing outer race fault diagnosis. The envelopes and SESs of the band-pass filtered signals obtained from six selected frequency bands are displayed in Figs. 15 and 16, respectively.
The repetitive impulse features can be clearly observed in Fig. 15a-f, and the outer race fault characteristic frequency f o and its first six harmonics can be easily detected in Fig. 16a-d and f. These results indicate that five frequency bands (4,800,3,200), (5,600, 1,600), (4,800, 1,067), (5,067,533) and (5,333,2,133) contain repetitive impulse features caused by bearing outer race fault. Because the frequency bands (5,600, 1,600), (4,800, 1,067) and (5,067,533) have relatively narrow bandwidths, the SESs of their corresponding filtered signals exhibit better fault diagnosis effects. In addition, IGIs with p = 0.1 and 6 fail to effectively identify the fault-related resonance frequency bands, which may be caused by the weaker repetitive transient discriminability of IGI with p = 0.1 and the susceptibility to the noise of IGI with p = 6. These results verify the effectiveness of IGIs in fault diagnosis of axle-box bearing outer race and also show that the power exponent of IGIs should not be too large or too small in the application.

B. BEARING ROLLING ELEMENT FAULT
The vibration signal of the rolling element fault bearing collected at a constant speed of 885 rpm is used to further validate the proposed method, as displayed in Fig. 17a. The repetitive impulse features induced by the rolling element fault are buried in the strong background noise. Although the harmonics 2f b , 3f b and 6f b can be observed from SES shown in Fig. 17c, the fault characteristic frequency of the bearing rolling element (f b = 48.15 Hz) and other harmonics are not obvious, resulting in the inability to accurately judge the health status of the axle-box bearing.
To extract the fault impulse features of bearing rolling element, the IGIgram using IGIs with p = 0.1, 0.3, 0.5, 0.8, 2, 3, 4 and 6, geometric infogram and its improved methods using Kurt, L2/L1, HI, GI and GGIs with a = 0.1, 0.3, 0.5, 0.8, 2, 3, 4 and 6 are applied to the vibration signal shown in Fig. 17a. The decomposition level of the entire frequency band is set to 4 for all methods. show the envelopes and SESs of the band-pass filtered signals obtained from the six selected frequency bands, respectively.
As shown in Fig. 18a-c and f, the repetitive impulse features can be clearly observed. Correspondingly, in Fig. 19a-c and f, the fault characteristic frequency of bearing rolling element f b and its first three harmonics can be easily identified, and the sidebands can also be detected. Thus, the results obtained from the band frequency bands (5,867, 1,067), (5,200, 800), (5,400, 400) and (5,600, 1,600) confirm the rolling element fault of the axle-box bearing. Furthermore, the SESs obtained from the frequency bands (5,867, 1,067) and (5,600, 1,600) achieve better effects than that of the other two frequency bands. The above results further verify the effectiveness of IGIs in IFB identification and bearing fault diagnosis.

VI. COMPARISON WITH LOG-CYCLIGRAM
To further illustrate the performance of the proposed method in bearing fault diagnosis, a state-of-the-art    envelope analysis method, LC [12], is applied to the simulation and experimental signals in Sections IV and V for comparison. The decomposition level of the entire frequency band is also set to 4. The bearing fault characteristic frequency calculated from the nominal speed and bearing geometric parameters is used as an input parameter of the LC method to guide the frequency band selection. Fig. 20 shows the frequency bands selected by the LC method on the simulation signal in Fig. 9a and experimental signals in Figs. 14a, and 17a, which are (5,000, 400), (4,600, 400) and (5,000, 400) respectively. Figs. 21 and 22 display the envelopes and SESs of the band-pass filtered signals for processing the simulation signal of bearing inner race fault, the experimental signal of bearing outer race fault and the experimental signal of bearing rolling element fault, respectively. As shown in Fig. 21, the repetitive transient features can be observed in the envelopes of the band-pass filtered signals. In Fig. 22, the bearing fault characteristic frequencies and their harmonics (indicated by the red dotted lines) can be detected in the SESs of the band-pass filtered signals. These results show that the LC method effectively extracts the transient features related to different bearing faults. Nevertheless, only the rolling element fault characteristic frequency 1f b and its harmonic 2f b can be clearly observed in Fig. 22c and the other harmonics are not prominent, which seems to be slightly inferior to the results shown in Fig. 19c, and f, where 3f b and even other harmonics can be detected. In addition, it should be pointed out that IGIgram can adaptively extract fault-related transient features, unlike the LC method which aims to extract repetitive transient features with a specified characteristic frequency. This shows that the IGIgram with appropriate power exponent p can achieve a fault diagnosis effect similar to the LC method and has advantages in the adaptive extraction of bearing fault features.

VII. CONCLUSIONS
In this paper, a generalization method of the classic GI is developed from the perspective of the quasi-arithmetic mean generator, and IGIs are proposed as new statistical indicators for sparse quantification. Further, the IFBI method that uses IGI as the fault information metric, named IGIgram, is introduced for envelope analysis-based bearing fault diagnosis. Simulations and experiments are conducted to verify the performance of IGIs and IGIgram against the typical sparsity measures. The following important conclusions can be drawn: (1) The proposed generalization method of classic GI is reasonable and effective. IGIs can monotonically quantify the sparsity of the signal and can achieve strong repetitive transient discriminability under noise contamination.
(2) The IGIgram method using IGI with appropriate power exponent can accurately discriminate IFB of the vibration signals to extract repetitive transients for bearing fault diagnosis.
(3) Compared with Kurt, L2/L1 and GI, IGIs with power exponents p = 2, 3 and 4 exhibit high random transient resistibility and strong repetitive transient discriminability simultaneously, therefore they are recommended for applications.
In future work, the performance comparison of IGIgram and other targeted IFBI methods in rotating machinery fault diagnosis is worth investigating. The fault diagnosis performance of IGIgram based on the log-envelope spectrum is also worth exploring. Furthermore, IGIs can be used to develop machine fault feature extraction methods and condition monitoring methods.