Available via license: CC BY 3.0

Content may be subject to copyright.

Fault Detection via Sparsity-based Blind Filtering on Experimental

Vibration Signals

Kayacan Kestel1, Cédric Peeters2, Jérôme Antoni3, and Jan Helsen4

1,2,4 Vrije Universiteit Brussel - VUB, Department of Applied Mechanics, Elsene, Brussels, 1050, Belgium

kayacan.kestel@vub.be

cedric.peeters@vub.be

jan.helsen@vub.be

3Univ Lyon, INSA Lyon, LVA, EA677, 69621 Villeurbanne, France

jerome.antoni@insa-lyon.fr

ABSTRACT

The detection of incipient rolling element bearing faults is a

challenging task since the impulsive pattern of bearing faults

often fades into the noise. Moreover, tracking the health con-

ditions of rotating machinery generally requires the character-

istic frequencies of the components of interest, which can be

a cumbersome constraint for large industrial applications be-

cause of the extensive number of machine components. One

recent method proposed in literature addresses these difﬁcul-

ties by aiming to increase the sparsity of the squared enve-

lope spectrum of the vibration signal via blind ﬁltering. As

the name indicates, this method requires no prior knowledge

about the machine. Sparsity measures of Hoyer index, l2

l1-

norm, and spectral negentropy are optimized in the blind

ﬁltering approach using generalized Rayleigh quotient iter-

ation. Even though the proposed method has demonstrated

a promising performance, it has only been applied to vibra-

tion signals of an academic experimental test rig. This pa-

per focuses on the real-world performance of the sparsity-

based blind ﬁltering approach on a complex industrial ma-

chine. One of the challenges is to ensure the numerical sta-

bility and the convergence of the generalized Rayleigh quo-

tient optimization. Enhancements are thus made by identify-

ing a quasi-optimal ﬁlter parameter range within which blind

ﬁltering tackles these issues. Finally, ﬁltering is applied to

certain frequency ranges in order to prevent the blind ﬁltering

optimization from getting skewed by dominant determinis-

tic healthy signal content. The outcomes prove that sparsity-

based blind ﬁlters are effective in tracking rolling element

bearing faults on real-world rotating machinery without any

prior knowledge of characteristic frequencies.

Kayacan Kestel et al. This is an open-access article distributed under the

terms of the Creative Commons Attribution 3.0 United States License, which

permits unrestricted use, distribution, and reproduction in any medium, pro-

vided the original author and source are credited.

1. INTRODUCTION

Early detection of the anomalous behaviour of rotating ma-

chinery has drawn signiﬁcant attention, since in large indus-

trial applications, maintenance and downtime costs can add

up to substantial amounts (Lu, Li, Wu, & Yang, 2009). Fur-

thermore, the complexity of rotating machinery has rapidly

increased thanks to technological developments in the recent

years. Accordingly, such machines are comprised of an im-

mense amount of components. Hence, it might complicate

keeping track of all the kinematic information of every com-

ponent. However, monitoring the health conditions of rotat-

ing machinery, in general, requires the knowledge of the char-

acteristic frequencies of dynamic components such as bear-

ings, shafts or gears. Thus, in the case of the lack or the

paucity of the kinematic information about the machine, fault

detection algorithms which are capable of functioning blindly

are needed.

Blind approaches that require no a-priori knowledge about the

machine kinematics have already been employed for prognos-

tic and diagnostic purposes. A basic form of blind approaches

is monitoring the statistical indicators of the vibration signal

in the time domain, examples include the root-mean-square

or the kurtosis of the vibration amplitude. On the other hand,

time-domain indicators do not provide information regarding

the type of the faulty component. Furthermore, assessing the

health status based on statistical indicators may be misleading

as they are sensitive to operating conditions such as shaft rota-

tion speed or load. A more advanced example of tracking the

time waveform statistics is Minimum Entropy Deconvolution

(MED) ﬁltering (Wiggins, 1978), which aims to maximize the

kurtosis of the time waveform. Various attempts to improve

MED have been presented in literature. The higher order mo-

ments than the fourth order, which is linked to kurtosis, of

the vibration signal are implemented by (Gray, 1979) with

1

ANN UAL CO NF ERE NC E OF T HE PROGNOSTICS AND HEA LTH MANAGEMENT SOCI ET Y 2021

MED approach. An enhanced way of estimating kurtosis is

multi-point kurtosis proposed by (McDonald & Zhao, 2017)

and a new indicator based on Jarque-Bera statistics is also

utilized as a detection measure on the time-domain signal via

blind deconvolution (Obuchowski, Zimroz, & Wyłoma´

nska,

2016). Recently new blind methods tend to exploit cyclosta-

tionarity rather than the time-domain statistics. Particularly

the detection of incipient roller bearing faults, which consti-

tutes the majority of the machine faults (Graney & Starry,

2012), entails scrutinizing the stochastic nature of roller bear-

ing impacts as their impulsive pattern demonstrates a sec-

ond order cyclostationary behavior. Therefore, several stud-

ies have focused on utilizing this content of the signal for

blind approaches. A novel design of a blind deconvolution

ﬁlter maximizes the cyclostationary content of the signal by

exploiting the generalized Rayleigh quotient (Buzzoni, An-

toni, & D'Elia, 2018), albeit that it requires a-priori knowl-

edge of the machine components. While this paper does not

aim to include an exhaustive literature survey, the aforemen-

tioned studies are a summary of the blind approaches that

have been employed for fault detection. (Peeters, Antoni, &

Helsen, 2020) proposed a blind ﬁltering method which max-

imizes the sparsity of the squared envelope spectrum (SES)

of the vibration signal. A brief introduction to the concept of

sparsity and sparsity measures is laid out in the next section.

The present study focuses on the applicability of the proposed

method (Peeters et al., 2020) to vibration signals measured

on complex industrial rotational machinery. The real-world

measurements from a complex industrial gearbox are used for

investigating the developed methods. Moreover, a parameter

study is performed to obtain a quasi-optimal ﬁlter parameter

space within which blind ﬁlters are numerically stable and

function in a time-efﬁcient way. The theory of the blind ﬁl-

tering method is laid out in the second section, along with

a brief derivation of the blind ﬁlter equations revealing the

generalized Rayleigh quotient. In the third section, results

demonstrating the parameter study and the performance of

blind ﬁltering to diagnose incipient roller bearing fault de-

tection are shown and discussed. The last section stresses

that blind ﬁltering methods can be used as an effective tool

in large industrial applications as a health monitoring tool for

cases where knowledge of characteristic frequency is lacking

or minimal.

2. METHODOLOGY

(Peeters et al., 2020) already stated that in order to increase

the sparsity of the SES, the vibration signal is ﬁltered to max-

imize the value of the sparsity measure. Three sparsity mea-

sures are chosen based on the criteria discussed by (Hurley &

Rickard, 2009) and based on their mathematical convenience

for the derivation of the blind ﬁlters. Three sparsity measures

discussed in (Peeters et al., 2020), namely l2

l1-norm, Hoyer

index and spectral negentropy, are used in this study.

2.1. Brief Introduction to Sparsity Measures

A signal representation can be considered sparse when a lim-

ited number of samples contains the majority of the energy

(Hurley & Rickard, 2009). While in this study the concept of

sparsity is utilized with a particular interest of signal process-

ing for vibration based condition monitoring of rotating ma-

chinery, it also can be applied in a variety of other domains,

such as oceanic engineering (Li & Preisig, 2007), image pro-

cessing (Krishnan, Tay, & Fergus, 2011) or medical imaging

(Leung et al., 2008). The concept of sparsity in this study is

utilized in a way to maximize the sparseness of the SES of the

blind ﬁltered signal, with the aim of detecting the presence of

peaks associated with a fault in the squared envelope spectra.

Envelope analysis is one of the most widely used techniques

to detect rolling element bearing faults (Abboud, Antoni,

Sieg-Zieba, & Eltabach, 2017). The envelope spectrum can

be obtained by estimating the spectrum of the signal enve-

lope. It is proven that monitoring the squared envelope spec-

trum of a signal is a more effective way of detecting rolling

element bearing faults compared to the non-squared enve-

lope spectrum (Ho & Randall, 2000), particularly for signals

whose envelope spectrum has a signal-to-noise ratio which is

more than unity. The amplitude of the analytic signal results

in the envelope of the signal, and the analytic signal can be

simply formed by summing the signal itself and its Hilbert

transform (Ho & Randall, 2000). However, in this study, to

reduce the complexity of the mathematical derivations, we

estimated the squared envelope spectrum by multiplying the

signal spectrum with its conjugate. Accordingly, following

derivations are made.

In order to increase the sparsity measures of interest of a noisy

signal x, it is convolved with a ﬁlter hto estimate the ideal

input s(Peeters et al., 2020):

s=x∗h(1)

The vector and matrix quantities are represented in bold char-

acters, in order to distinguish them from the scalar ones. In

matrix form, the convolution operation can be written as:

s=Xh (2)

and the elements in the convolution operation correspond to:

s0

.

.

.

sL−1

=

xN−1. . . x0

.

.

.....

.

.

xL+N−2. . . xL−1

h0

.

.

.

hN−1

(3)

where lengths of sand hare Land N, respectively. Accord-

ingly the squared envelope xcan be deﬁned as:

2

ANN UAL CO NF ERE NC E OF T HE PROGNOSTICS AND HEA LTH MANAGEMENT SOCI ET Y 2021

x=|Xh|2=diag(sH)X h (4)

with diag(sH)being diagonal matrix of the Hermitian trans-

pose of s. Finally, squared envelope spectrum is estimated as

the Fourier transform of x:

Ex=Fdiag(sH)X h (5)

and the Fourier matrix Fis:

1 1 . . . 1. . . 1

1ω . . . ωn. . . ωL−1

.

.

..

.

..

.

..

.

..

.

..

.

.

1ωk. . . ωkn . . . ωk(L−1)

.

.

..

.

..

.

..

.

..

.

..

.

.

1ωK−1. . . ω(K−1)n. . . ω(K−1)(L−1)

(6)

with the basis function being ω=e−2πj/L ,n= 0, .. L −1

and k= 0, .. K −1. Accordingly, Fourier matrix Fhas the

dimensions of (K, L)and Kis the number of frequency in-

dex of the Fourier spectrum. An endeavour is made to ﬁnd an

optimum blind ﬁlter h, which maximizes the sparsity mea-

sures of the quantity Ex.

2.1.1. Sparsity Measures

One of the most well-known sparsity measures is l2

l1-norm.

The expression to estimate this norm is:

l2

l1

=qPL−1

n=0 |Ex(n)|2

PL−1

n=0 |Ex(n)|(7)

where L is the number of samples in the SES, i.e. signal

length.

The second measure employed to quantify the sparsity is

Hoyer index (HI) (Hoyer, 2004), which is essentially a nor-

malized version of l2

l1-norm (Hurley & Rickard, 2009), and

can be expressed as:

HI =√L−l1

l2

√L−1(8)

A shortcoming of the l2

l1-norm is being unbounded, hence, it

tends to result in different values for signals with different

lengths but with the same sparsity. On the other hand, Hoyer

index is bounded between unity and zero. For signals where

the energy is accumulated into a single point, the Hoyer in-

dex results in unity, and it converges to zero when the energy

content becomes more equally distributed (Hoyer, 2004).

The third sparsity measure is spectral negentropy. Based on

the deﬁnition of entropy, one can deﬁne the spectral entropy

of a signal as the probability distribution of its power spec-

trum. Accordingly, the lowest entropy arises for the cases

in which the energy of the signal accumulates into a single

impulse (Antoni, 2016). In other words, the lower the value

of spectral entropy of a signal, the more sparse the signal is,

which the makes negative of the spectral entropy a plausible

candidate to measure sparsity. This deﬁnition is already pro-

posed in literature as the spectral negentropy (Antoni, 2016)

and it is expressed for the SES as:

∆IE=|Ex|2

h|Ex|2iln |Ex|2

h|Ex|2i (9)

2.1.2. Generalized Rayleigh Quotient Derivation

The optimal ﬁlter coefﬁcient estimation is achieved employ-

ing generalized Rayleigh quotient iteration. A brief deriva-

tion of the blind ﬁlter equations is presented for the l2

l1-norm.

Rewriting the equation Eq. 7 in vector notation results in:

l2

l1

=pExHEx

ExHdiag(1

|Ex|)Ex

(10)

Further manipulation of both numerator and denominator of

Eq. 10 gives:

l2

l1

=

ExHdiag(1

√Ex

HEx

)Ex

ExHdiag(1

|Ex|)Ex

(11)

and the details of which are presented in (Peeters et al., 2020).

Plugging Eq. 5 into Eq. 11 results in:

l2

l1

=hHXHdiag(s)FHA F diag(sH)Xh

hHXHdiag(s)FHB F diag(sH)X h (12)

where Aand Bcorrespond to:

A=diag(1

pExHEx

)(13)

and

B=diag(1

|Ex|)(14)

respectively. Equation 12 reveals the generalized Rayleigh

quotient (Horn & Johnson, 1985). Thus, Eq. 12 can be sim-

pliﬁed utilizing the deﬁnition of that as:

3

ANN UAL CO NF ERE NC E OF T HE PROGNOSTICS AND HEA LTH MANAGEMENT SOCI ET Y 2021

l2

l1

=hHRXA h

hHRXB h(15)

with:

RXA =XHdiag(s)FHA F diag(sH)X(16)

and

RXB =XHdiag(s)FHB F diag(sH)X(17)

As the proposed approach in (Peeters et al., 2020) is to in-

crease the sparsity of the squared envelope spectra of the

blind ﬁltered signal, an iterative solution is performed to

solve for hto maximize l2

l1-norm in Eq. 15. With regards

to the maximization, the maximum value of the generalized

Rayleigh quotient with respect to his equivalent to its largest

eigenvalue and corresponding eigenvector. Therefore, the as-

sociated generalized eigenvalue problem with the Rayleigh

quotient deﬁned in Eq. 15 can be written as:

RXA h=RXB hλ(18)

and it is iteratively solved to estimate the optimal ﬁlter coefﬁ-

cients which are inherently equal to eigenvector of the prob-

lem. For the detailed explanation of the iteration steps and

the derivation of the two remaining sparsity measures, read-

ers can refer to (Peeters et al., 2020).

One of the strong aspects of generalized Rayleigh quotient it-

eration is its drastic convergence rate (Parlett, 1974), as long

as the numerical stability is satisﬁed. Thus, as a part of the

present study, numerical stability is also investigated by per-

forming a broad parametric study. The sensitivity of the con-

vergence characteristics of the iteration process to ﬁlter length

and its initialization is also scrutinized.

2.1.3. Filter Initialization

While it is mentioned that the convergence rate of Rayleigh

quotient iteration is rapid, an initial guess considerably far

from the solution domain of the ﬁlter may result in diver-

gence. In this study, iterations initialized with two different

approaches are also compared. The ﬁlters are initialized by

either a differentiation ﬁlter or an autoregressive (AR) model.

The differentiation ﬁlter is basically comprised of zeros ex-

cept for the second and the fourth coefﬁcients which are set to

be 1and −1, respectively, regardless of the ﬁlter length. The

differentiation ﬁlter is further normalized prior to the iteration

process. The coefﬁcients of AR model, on the other hand, are

estimated using Levinson-Durbin recursion (Franke, 1985).

2.2. Experimental Problem Deﬁnition

The vibration signals investigated in this study were sampled

at 40 kHz for 2seconds at every 10 minutes and a dataset

containing 73 measurements was obtained. The average rota-

tional speed of the high-speed shaft was 279 Hz. In general

angular resampling is necessary for a proper vibration based

condition monitoring of a rotating machine. The angular re-

sampling requires the knowledge of the instantaneous speed

of the shaft so that the data can be transformed to the angu-

lar domain in order to compensate for the speed variations

(Peeters et al., 2019). Nevertheless, the present dataset was

sampled under negligible speed variations, hence the angular

resampling is not needed. Accordingly, the signal spectrum

is dominated by pronounced shaft harmonics. Another po-

tential pre-processing method to improve the effectiveness of

blind ﬁltering is the deterministic content removal. The high-

energy deterministic content protrudes in the squared enve-

lope spectrum and might mask the bearing fault signature.

Since the localized bearing faults manifest themselves as a

stochastic process, the energy level of which are lower in the

squared envelope spectrum compared to deterministic con-

tent originated from the other machine components (Antoni

& Randall, 2003). Thus, in order to ensure that the sparsity

measure of the blind ﬁltered signal is not skewed by the high-

energy harmonics, the deterministic content of the signal can

be removed prior to blind ﬁltering using several techniques

discussed in literature, i.e. cepstral editing or discrete-random

separation (Peeters et al., 2020). However, the signals are not

pre-whitened in this study, but instead bandwidths to which

blind ﬁltering is applied are adjusted in such a way as to ex-

clude the prominent amplitudes related to the shaft harmonics

in squared envelope spectrum. This step is explained in the

next section.

The last measurement in the data set corresponds the one

where the machine has failed. The damaged inner ring of

the rolling element bearing can be seen in Fig. 1. The ma-

chine is stressed at full capacity for the duration of the exper-

iment which induces high loads on the bearings and signiﬁ-

cantly accelerates the bearing degradation. The evolution of

the bearing fault in the zoomed squared envelope spectra is

discernable in Fig. 2 with an increase in the amplitude at the

ballpass frequency inner race (BPFI) order of 8.29. Approx-

imately after the measurement 30, which is indicated with

the vertical dashed line, the SES amplitude around order 8.29

becomes distinct, which can be easily monitored by tracking

the BPFI order. Two basic statistical indicators estimated us-

ing time-domain waveform of the signal are demonstrated in

Fig. 3. An increasing trend for both the root-mean-square and

the kurtosis of the signal is observed from measurement 43

onwards, which is an indication of a potential bearing fault.

While these statistical indicators can be considered ’blind’,

they do not include any ﬁlter optimization nor do they try to

4

ANN UAL CO NF ERE NC E OF T HE PROGNOSTICS AND HEA LTH MANAGEMENT SOCI ET Y 2021

embed engineering knowledge of the vibration signal into the

analysis.

Figure 1. Picture of the damaged inner ring of the rolling

element bearing.

Figure 2. The evolution of the bearing fault in the squared

envelope spectra of the raw signals.

3. RES ULTS

The experimental results are presented in two sections. The

ﬁrst section discusses the salient points of the parameter

study. This study aims to ﬁnd quasi-optimal parameter ranges

where the blind ﬁltering approach can provide robust results.

The second section presents the performance of the three

sparsity measures.

3.1. Parametric Study

There exist three main parameter settings which may change

the numerical stability or the convergence rate of the blind ﬁl-

tering method, which are the ﬁlter length,ﬁlter initialization,

and frequency range that is to be ﬁltered.

The decision of the length of the ﬁlter is signiﬁcant as it af-

fects both the computation time and the convergence of the

506

.XUWRVLV

0HDVXUHPHQW1R>@

506/HYHO

.XUWRVLV/HYHO

Figure 3. Evolutions of the root-mean-square and the kurtosis

of the time waveform of the raw signal.

iteration process. While a longer ﬁlter is capable of resolv-

ing the ﬁner frequency content compared to a shorter one, the

latter tends to improve the convergence. This is due to the

fact that longer ﬁlters inherently require more coefﬁcients to

be optimized, accordingly, the global minimum of a larger

domain may never be achieved. Figure 4 displays the evolu-

tion of the Hoyer indexes for different ﬁlter lengths N. The

Hoyer index of the signals that are blind ﬁltered by the ﬁlters

with the lengths of 5,10,20,50 and 100 are compared. Even

though the evolutions of the indicator estimated for different

ﬁlter lengths do not demonstrate any promising trend for the

fault detection means, the curves do not exhibit any arbitrary

outlier peaks or drops, which indicates that all ﬁve lengths

provide a numerically stable solution.

1

1

1

1

1

0HDVXUHPHQW1R>@

+R\HULQGH[

Figure 4. The evolution of Hoyer index of the SES of the

signals blind ﬁltered using different ﬁlter lengths.

5

ANN UAL CO NF ERE NC E OF T HE PROGNOSTICS AND HEA LTH MANAGEMENT SOCI ET Y 2021

Like in any iterative solution, initial guess can play an impor-

tant role for both the convergence itself and its rate. How-

ever, Fig. 5 reveals that there is no considerable distinction

between the results obtained for the two different ﬁlter ini-

tialization methods. Despite there being a slight discrepancy

between the two curves in Fig. 5, the general trend in both

curves is nearly identical. Therefore, both initialization meth-

ods provide a stable result. Moreover, no distinction is de-

tected for the computation time of the iteration processes ini-

tiated with these two settings. One may also ﬁnd out that the

line corresponding to the ﬁlter length 10 in Fig. 4 is identical

to the AR line in Fig. 5, since these two curves processed

with the same parameter settings. Hence, numerical stability

of the Rayleigh quotient iteration is achieved for ﬁlters ini-

tialized with both the differentiation ﬁlter and the AR model.

GLIIHUHQWLDWLRQ

$5

0HDVXUHPHQW1R>@

+R\HULQGH[

Figure 5. The evolution of Hoyer index of the SES of the

signals blind ﬁltered using different ﬁlter initializations.

The last setting mentioned is the frequency range to which

the blind ﬁlter is applied. The optimization can be narrowed

down to only take into account the sparsity of a certain fre-

quency band in the squared envelope spectrum. This is an

important setting because of the fact that narrower frequency

band in the SES means shorter convergence time whereas it

also poses the risk of excluding the frequency content of in-

terest which is assumed to be unknown. In order to prevent

such a case, the initial attempts started by employing the full

available frequency range (from zero to Nyquist). The up-

per limit of the frequency range gradually decreased to 10,5,

and 3kHz and signals ﬁltered within these frequency ranges

are displayed in Fig. 6. It can be stated that frequency range

has no effect on the numerical stability of the Rayleigh quo-

tient optimization for reasonably wide frequency bands. The

numerical stability is not studied for the very narrow fre-

quency band ﬁltering, because it offers no use for the blind

approaches in which very narrow band ﬁltering might jeop-

ardize fault detection by excluding the faulty frequency con-

tent. Nonetheless, as expected, computation time decreases

as the frequency range shrinks because it contains less points

to process. Having this in mind, applying blind ﬁltering to

wisely selected narrow bands appears to be an effective so-

lution, especially for the industrial applications where execu-

tion time of the operations is important. On the other hand,

except for 1−3000 Hz line, none of the other lines provides

a promising trend in terms of fault detection in Fig. 6. The

line of 1−3000 Hz demonstrates an increasing trend from

measurement 50 onwards, albeit that increment is consider-

able small. Also one can realize that the curve for the full

frequency range is identical to the mentioned curve in Figs. 4

and 5. Hence, results presented in Fig. 6 are obtained using

the same parameter settings.

+]

+]

+]

+]

0HDVXUHPHQW1R>@

+R\HULQGH[

Figure 6. The evolution of Hoyer index of the SES of the

signals blind ﬁltered using different frequency bands.

As the conclusion of this parameter study, it is observed that

while longer ﬁlters might result in non-convergence and re-

quire considerably long computation time, ﬁlter initialization

and the frequency range settings do not impose a signiﬁ-

cant effect on the generalized Rayleigh quotient maximiza-

tion process with this regard. The reason as to why longer sig-

nal may complicate the optimization process, as mentioned

above, is because the increase in the length of the ﬁlter means

more constants to be estimated. Therefore, even though con-

vergence of the optimization may be satisﬁed, trend of the

signals’ sparsity measure becomes quite noisy for longer ﬁl-

ters, which is undesirable in terms of alarming point of view.

Considering the outcome the parametric study, for the rest of

the study blind ﬁlters are initialized with AR method for the

ﬁlter length of 10.

3.2. Fault Detection

Now that it is proven that there exists a quasi-optimal pa-

rameter settings range for a stable numerical convergence,

6

ANN UAL CO NF ERE NC E OF T HE PROGNOSTICS AND HEA LTH MANAGEMENT SOCI ET Y 2021

the rest of the endeavour is made to employ sparsity-based

blind ﬁltering approach to diagnose the fault from the exper-

imental vibration signal. To do so, evolutions of the spar-

sity measures must be tracked. Referring back to the Fig. 6,

none of the trends displays an increase to indicate any change

in the sparsity of the squared envelope spectrum, albeit that

these curves are achieved by blind ﬁltering with the quasi-

optimal parameter settings. Reminding the discussion about

deterministic content removal, a potential culprit for the un-

derperformance of the blind ﬁltering approach might be the

presence of strong shaft harmonics masking the bearing fault

signature in the squared envelope spectrum. As mentioned

above, signal spectrum is dominated with the shaft harmon-

ics which may skew the sparseness of the SES as a result of

the blind ﬁltering. It is known that the experimental vibra-

tion signals of interest are sampled with a negligible varia-

tion of the shaft speed. Therefore, given the average speed of

the high-speed shaft, the frequency span is divided into (fre-

quency) bands enclosed by the shaft harmonics. In order to

ensure that the sparsity measure is not skewed due to the high

amplitude frequency bins adjacent to the shaft harmonics, fre-

quency bins in the range of ±3% of the shaft harmonics are

also excluded. The exception is made for the 0th harmonic,

as it contains no useful information for diagnostic purposes.

Thus, setting the lower and the upper bounds of the frequency

bands with the shaft harmonics and ±3% off-set, Table 1 is

generated. In each row, the lower and the upper bounds of the

frequency span are shown, along with the grey row which is

where the BPFI information is embedded.

Table 1. The frequency bands within which the signals are

blind ﬁltered.

No Harmonics Lower [Hz] Upper [Hz]

1 0 - 1 1 271

2 1 - 2 287 550

3 2 - 3 566 829

4 3 - 4 845 1108

5 4 - 5 1124 1387

6 5 - 6 1403 1666

7 6 - 7 1682 1945

8 7 - 8 1961 2224

9 8 - 9 2240 2503

10 9 - 10 2519 2782

In Figs. 7, 8, and 9, the evolutions of sparsity measures of

the spectral negentropy, Hoyer index and the l2

l1-norm are

demonstrated, respectively. The ﬁgures include the sparsity

measures estimated on the different frequency bands of the

squared envelope spectrum. Hence, in each ﬁgure, curves

correspond to the information contained in the frequency bins

enclosed by shaft harmonics of 7−8,8−9,9−10 as well as

in the frequency band of 1−3000 Hz, which is also demon-

strated in Fig. 6, are shown. The purpose of presenting

the latter is to emphasize the difference in the evolution of

trends. In order to ease the representation of Figs. 7, 8, and

9, lines corresponding to the sparsity measures obtained by

ﬁltering the frequency bands shown in Table 1 are coloured

with black, whereas grey line represents the results for ﬁlter-

ing the frequency band of 1−3000 Hz.

Figure 7 displays the trend of the spectral negentropy of the

squared envelope spectrum of the blind ﬁltered signals. Even

at ﬁrst glance, a sudden increase in the trend after a plateau

phase can be observed for the black lines. The effective-

ness of the proposed approach of shaft harmonic exclusion

is clearly more pronounced compared to the trend of the solid

grey line which corresponds to ﬁltered signal frequency con-

tent within 1−3000 Hz. As mentioned in the previous sec-

tions, the impulsive pattern of incipient rolling element bear-

ing fault results in the increase in the amplitude in the related

frequency bin, which also increases the sparsity. Given that

the spectral negentropy increases almost 4 times its original

value, it forms a clear indication of a potential bearing fault.

On the other hand, the solid grey curve demonstrates a slight

increase after the measurement 50, which may not be directly

linked to a faulty component of the machine as the increase

is not clear compared to the mean of the trend of the earlier

measurements. This is an important observation particularly

for the industrial applications where the condition monitor-

ing is performed autonomously and the detection of a fault

requires clear elevating trend exceeding a threshold. Hence,

such minuscule increase in the trend may not be considered

as a clear indication of a fault.

+]

+]

+]

+]

0HDVXUHPHQW1R>@

6SHFWUDOQHJHQWURS\

Figure 7. The evolution of the spectral negentropy of the SES

of the blind ﬁltered signals.

Similar comments can be made for the trends of Hoyer index

and l2

l1-norm shown in Figs. 8 and 9, respectively. Consid-

ering the Hoyer index, the indicator almost doubles from the

27th measurement to the 40th one for the solid black curve.

Although the faulty signal content is embedded in the fre-

7

ANN UAL CO NF ERE NC E OF T HE PROGNOSTICS AND HEA LTH MANAGEMENT SOCI ET Y 2021

quency range of 2240 −2503 Hz, as highlighted in Table

1, trends with clear indication of the fault are present in the

curves of 1961−2224 Hz and 2519−2782 Hz as well. On the

other hand, blind ﬁltering the signals for the frequency bands

numbered in Table 1 from 1 to 3 does not demonstrate any

clear increase in the sparsity measures. The Hoyer index evo-

lutions of signals blind ﬁltered for the frequency bandwidths

number 2 and 3 are depicted in Fig. 10, both of which are

ﬂat, thus, no indication of a fault. Similarly, for the frequency

band of number 4 in Table 1, the onset of the surge in Hoyer

index is delayed relative to the that of the line corresponding

to the frequency range 2240 −2503 Hz, as shown in Fig. 10.

Filtering the signal within the frequency band shown in row

6 in Table 1 also demonstrates a similar performance as the

cases presented with the black lines in the ﬁgures, yet it is not

shown for the sake of the clarity of the ﬁgures. Therefore, the

indication of the faulty bearing is apparent in the trends of the

sparsity measures estimated for several frequency bands.

+]

+]

+]

+]

0HDVXUHPHQW1R>@

+R\HULQGH[

Figure 8. The evolution of the Hoyer index of the SES of the

blind ﬁltered signals.

A ﬁnal comparison is also made between the condition mon-

itoring methods where the characteristic frequency knowl-

edge is utilized and the blind ﬁltering approach. Figure 11

demonstrates evolutions of the sum of amplitudes (SoA) of

the squared envelope spectrum around the BPFI order 8.29

and of the Hoyer index for the frequency range number 9 in

Table 1 where the faulty frequency content is contained. Al-

beit that it is not a clear-cut distinction, the earliest sudden

increase in SoA occurs later than that in Hoyer index does so.

This may imply that blind ﬁltering approach to maximize the

sparsity of the squared envelope spectrum performs as effec-

tive as classical condition monitoring tools based on tracking

the characteristic frequencies, if not outperforms.

While the focus of this study is not aiming to compare the

performances of the sparsity measures, for an industrial au-

+]

+]

+]

+]

0HDVXUHPHQW1R>@

//1RUP

Figure 9. The evolution of the l2

l1-norm of the SES of the blind

ﬁltered signals.

tonomous condition monitoring application, Hoyer index can

be said to be the most useful, since it is scaled in between zero

to unity. For the vibration signal dataset investigated in this

study, the last measurement where the Hoyer index reached a

value around 0.8 is already the one where the machine failed.

Considering that, further studies may be performed to set a

threshold to assess the severity of the damage and to warn

the end-user of the potential degradation of a component of

interest.

4. CONCLUSION

This paper investigated the performance of sparsity-based

blind ﬁltering approaches on experimental data. It is proven

that, with a known the average or the instantaneous shaft

speed, a blind ﬁltering approach is capable of detecting the

incipient rolling element bearing fault of a complex industrial

rotating machine. Furthermore, results of the broad parame-

ter study stress that ﬁlter length is a signiﬁcant parameter that

inﬂuences the convergence characteristic of the iterative so-

lution that maximizes the generalized Rayleigh quotient. For

the given dataset, the conclusion can be drawn that ﬁlter ini-

tialization induces no effect on either the numerical stability

or the computation time of the iteration process. In summary,

the blind ﬁltering method can be used as an autonomous

health surveillance tool for complex industrial applications

in which exact characteristic frequencies of mechanical com-

ponents are lacking.

ACKNOWLEDGMENT

Kayacan Kestel, Cédric Peeters, and Jan Helsen received

funding from the Flemish Government (AI Research Pro-

gram). They would like to acknowledge FWO (Fonds Weten-

schappelijk Onderzoek) for their support through the post-

8

ANN UAL CO NF ERE NC E OF T HE PROGNOSTICS AND HEA LTH MANAGEMENT SOCI ET Y 2021

+] +]

+] +]

0HDVXUHPHQW1R>@

+R\HULQGH[

Figure 10. The comparison of the evolutions of the Hoyer

index of the SES of the blind ﬁltered signals for different fre-

quency bandwidths.

doctoral grant of Cédric Peeters (#1282221N). They would

also like to acknowledge FWO for the support through the

SBO Robustify project (S006119N).

NOMENCLATURE

AR autoregressive

hblind ﬁlter

sideal input

FFourier matrix

HI Hoyer index

MED minimum entropy deconvolution

xnoisy signal

∆IEspectral negentropy

xsquared envelope

Exsquared envelope spectrum

SES squared envelope spectrum

SoA sum of amplitudes

REFERENCES

Abboud, D., Antoni, J., Sieg-Zieba, S., & Eltabach, M.

(2017). Envelope analysis of rotating machine vibra-

tions in variable speed conditions: A comprehensive

treatment. Mechanical Systems and Signal Processing,

84, 200–226.

Antoni, J. (2016, jun). The infogram: Entropic evidence

of the signature of repetitive transients. Mechani-

cal Systems and Signal Processing,74, 73–94. doi:

10.1016/j.ymssp.2015.04.034

Antoni, J., & Randall, R. B. (2003, jun). A stochastic model

for simulation and diagnostics of rolling element bear-

ings with localized faults. Journal of Vibration and

Acoustics,125(3), 282-289.

+R\HU,QGH[

6R$

0HDVXUHPHQW1R>@

+R\HU,QGH[/HYHO

6XPRI$PSOLWXGHV

Figure 11. The comparison between the evolutions of Hoyer

index of the SES of the blind ﬁltered signals and the sum of

amplitudes of the SES of the raw signals in the vicinity of the

characteristic frequency.

Buzzoni, M., Antoni, J., & D'Elia, G. (2018, oct). Blind

deconvolution based on cyclostationarity maximization

and its application to fault identiﬁcation. Journal of

Sound and Vibration,432, 569-601.

Franke, J. (1985). A levinson-durbin recursion for

autoregressive-moving average processes. Biometrika,

72(3), 573–581.

Graney, B. P., & Starry, K. (2012). Rolling element bearing

analysis. Materials Evaluation,70(1), 78.

Gray, W. C. (1979). Variable norm deconvolution (Unpub-

lished doctoral dissertation). Stanford University Ph.

D. thesis.

Ho, D., & Randall, R. (2000). Optimisation of bearing di-

agnostic techniques using simulated and actual bearing

fault signals. Mechanical systems and signal process-

ing,14(5), 763–788.

Horn, R. A., & Johnson, R. (Eds.). (1985). Matrix analysis.

Cambridge University Press.

Hoyer, P. O. (2004). Non-negative matrix factorization with

sparseness constraints. Journal of machine learning re-

search,5(9).

Hurley, N., & Rickard, S. (2009). Comparing measures of

sparsity. IEEE Transactions on Information Theory,

55(10), 4723-4741.

Krishnan, D., Tay, T., & Fergus, R. (2011, jun). Blind de-

convolution using a normalized sparsity measure. In

CVPR 2011. IEEE.

Leung, K. Y. E., van Stralen, M., Nemes, A., Voormolen,

M. M., van Burken, G., Geleijnse, M. L., . . . Bosch,

J. G. (2008). Sparse registration for three-dimensional

stress echocardiography. IEEE Transactions on Medi-

cal Imaging,27(11), 1568-1579.

Li, W., & Preisig, J. C. (2007). Estimation of rapidly time-

9

ANN UAL CO NF ERE NC E OF T HE PROGNOSTICS AND HEA LTH MANAGEMENT SOCI ET Y 2021

varying sparse channels. IEEE Journal of Oceanic En-

gineering,32(4), 927-939.

Lu, B., Li, Y., Wu, X., & Yang, Z. (2009, jun). A review of

recent advances in wind turbine condition monitoring

and fault diagnosis. In 2009 ieee power electronics and

machines in wind applications.

McDonald, G. L., & Zhao, Q. (2017, jan). Multipoint optimal

minimum entropy deconvolution and convolution ﬁx:

Application to vibration fault detection. Mechanical

Systems and Signal Processing,82, 461-477.

Obuchowski, J., Zimroz, R., & Wyłoma´

nska, A. (2016, jun).

Blind equalization using combined skewness–kurtosis

criterion for gearbox vibration enhancement. Measure-

ment,88, 34-44.

Parlett, B. N. (1974). The rayleigh quotient iteration and

some generalizations for nonnormal matrices. Mathe-

matics of Computation,28(127), 679-693.

Peeters, C., Antoni, J., & Helsen, J. (2020, apr). Blind

ﬁlters based on envelope spectrum sparsity indicators

for bearing and gear vibration-based condition mon-

itoring. Mechanical Systems and Signal Processing,

138, 106556.

Peeters, C., Leclère, Q., Antoni, J., Lindahl, P., Donnal, J.,

Leeb, S., & Helsen, J. (2019, aug). Review and

comparison of tacholess instantaneous speed estima-

tion methods on experimental vibration data. Mechan-

ical Systems and Signal Processing,129, 407–436.

Wiggins, R. A. (1978). Minimum entropy deconvolution.

Geoexploration,16(1-2), 21-35.

10