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Fault detection via sparsity-based blind filtering on experimental vibration signals

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Abstract

Detection of bearing faults is a challenging task since the impulsive pattern of bearing faults often fades into the noise. Moreover, tracking the health conditions of rotating machinery generally requires the characteristic frequencies of the components of interest, which can be a cumbersome constraint for large industrial applications because of the extensive number of machine components. One recent method proposed in literature addresses these difficulties by aiming to increase the sparsity of the envelope spectrum of the vibration signal via blind filtering (Peeters. et al., 2020). As the name indicates, this method requires no prior knowledge about the machine. Sparsity measures like Hoyer index, l1/l2 norm, and spectral negentropy are optimized in the blind filtering approach using Generalized Rayleigh quotient iteration. Even though the proposed method has demonstrated a promising performance, it has only been applied to vibration signals of an academic experimental test rig. This paper focuses on the real-world performance of the sparsity-based blind filtering approach on a complex industrial machine. One of the challenges is to ensure the numerical stability and the convergence of the Generalized Rayleigh quotient optimization. Enhancements are thus made by identifying 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 deterministic healthy signal content. The outcome proves that sparsity-based blind filters are effective in tracking bearing faults on real-world rotating machinery without any prior knowledge of characteristic frequencies.
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
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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=xh(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
.
.
.
sL1
=
xN1. . . x0
.
.
.....
.
.
xL+N2. . . xL1
h0
.
.
.
hN1
(3)
where lengths of sand hare Land N, respectively. Accord-
ingly the squared envelope xcan be defined as:
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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. . . ωL1
.
.
..
.
..
.
..
.
..
.
..
.
.
1ωk. . . ωkn . . . ωk(L1)
.
.
..
.
..
.
..
.
..
.
..
.
.
1ωK1. . . ω(K1)n. . . ω(K1)(L1)
(6)
with the basis function being ω=e2πj/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
=qPL1
n=0 |Ex(n)|2
PL1
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 =Ll1
l2
L1(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:
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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
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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
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506
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0HDVXUHPHQW1R>@
506/HYHO
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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
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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.
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GLIIHUHQWLDWLRQ
$5
0HDVXUHPHQW1R>@
+R\HULQGH[
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 13000 Hz line, none of the other lines provides
a promising trend in terms of fault detection in Fig. 6. The
line of 13000 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.
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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,
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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 78,89,910 as well as
in the frequency band of 13000 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 13000 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 13000 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.
  
+]
+]
+]
+]
0HDVXUHPHQW1R>@
6SHFWUDOQHJHQWURS\
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 19612224 Hz and 25192782 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.
  




+]
+]
+]
+]
0HDVXUHPHQW1R>@
+R\HULQGH[
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-
  




+]
+]
+]
+]
0HDVXUHPHQW1R>@
//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
  




+] +]
+] +]
0HDVXUHPHQW1R>@
+R\HULQGH[
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$
0HDVXUHPHQW1R>@
+R\HU,QGH[/HYHO
6XPRI$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
... The Rayleigh quotient iteration benefits from its rapid convergence rate as long as the numerical stability is satisfied [11]. A study on the effect of the filter parameters on the Rayleigh quotient iteration demonstrates that filter initialization does not significantly influence the solution while its length is problem-dependent [12]. Therefore, in this study, filters are initialized with the simple difference filter. ...
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This study attempts to improve the performance of Generalized Likelihood Ratio Test-based indicators via blind filtering the of vibration signals. The key point is the optimization of the filter coefficients to maximize the indicator of interest. The filter coefficients are optimized through Rayleigh quotient iteration. The proposed method's performance and applicability are demonstrated on both simulated and real vibration signals measured on an experimental test rig. The outcome of the study shows that the Rayleigh quotient iteration is a potent tool for maximizing such complex condition monitoring indicators. Inspections over the filtered signals reveal that the optimal filters promote particular signal patterns linked to a bearing fault in vibration signals. The indicator estimated over the filtered signals is able to detect the bearing fault more robustly when compared to the raw signals.
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