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Mechanical Systems and Signal Processing 22 (2008) 520–541

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Blind extraction of a cyclostationary signal using reduced-rankcyclic regression—A unifying approach

Roger Boustany�, Jerome Antoni

Laboratory Roberval of Mechanics, Centre de Recherche de Royallieu, University of Technology of Compiegne, 60205 Compiegne, France

Received 2 September 2005; received in revised form 30 April 2007; accepted 20 September 2007

Available online 6 October 2007

Abstract

This paper addresses the issue of the blind extraction of a second-order cyclostationary source drowned by an unknown

number of interferences and additive noise. It first reviews two recently developed methods based respectively on a

subspace decomposition of the observed signals via their cyclic statistics and on multiple cyclic regression (MCR). It then

proposes a unifying and refined approach using reduced-rank cyclic regression (RRCR) which combines the respective

advantages of the two previous methods and suppresses their drawbacks. It also reveals that unlike the classical MCR

technique, the power of the additive noise at the output of RRCR does not depend neither on the number of frequency

shifts used in the regression nor on the number of available measured signals. This property is verified by means of

simulations where the behaviour of all the methods with respect to many parameters is compared. RRCR is finally applied

to the diagnostics of bearings and gears where it is shown to achieve a very good extraction of fault signatures.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Blind signal extraction; Blind source separation; Cyclostationary signals; Subspace decomposition; Multiple cyclic regression;

Reduced-rank cyclic regression; Bearing diagnostics; Gear diagnostics

1. Introduction

In the last two decades, several blind algorithms dedicated to the separation of mixtures of unobservedsignals (sources) have emerged [1,2]. The so-called blind source separation (BSS) algorithms usually rely on therestrictive assumptions of the knowledge of the number of sources present in the mixture and on their mutualstatistical independence—and thus uses independent component analysis (ICA) to recover them [3].

In the last few years, there have been some attempts to apply BSS to mechanical systems [4–6].Unfortunately, the task turned out to be more difficult than in communication, biomedical or other fields.This is due to many reasons. Among these, one can cite the fact that in such a complex environment, mutualstatistical independence may not apply and the number of sources is often large and unknown. Moreover, thecomplex convolutive mixture model is more appropriate to mechanical signals than the simple instantaneousone for which ICA was first dedicated.

e front matter r 2007 Elsevier Ltd. All rights reserved.

ssp.2007.09.014

ing author.

esses: roger.boustany@utc.fr (R. Boustany), jerome.antoni@utc.fr (J. Antoni).

ARTICLE IN PRESSR. Boustany, J. Antoni / Mechanical Systems and Signal Processing 22 (2008) 520–541 521

However, in many situations, the important information in the mixture is carried by only one component—the rest being interferences or even noise. This is for instance the case when one seeks the mechanical signatureof one particular fault in the system. This is a particular case of BSS where one is interested in separating onlyone signal of interest (SOI) and it will be referred to as blind signal extraction (BSE).

The issue of BSE consists of recovering one signal (or sequentially more than just one) using an a priori

knowledge on its properties. It proceeds by choosing a feature that favours this component and rejects all theother components present in the mixture. Such a feature may be stochastic as in [7], where signals are extractedfor instance in the order of absolute values of the generalised normalised kurtosis. Other possible extractioncriteria are the periodicity/non-stationarity of the SOI [8], or its cyclostationarity [9], etc.

BSE was recently applied to mechanical systems and in particular to rotating machines that usually generatevibration signals exhibiting cyclostationarity [10]. Indeed, the authors of [11] developed a multiple cyclicregression (MCR) technique that may be seen as an extension of the classical cyclic wiener filtering [12]. Thismethod proposes denoising the measured signal by decomposing it into its periodic, second-ordercyclostationary, and noise parts. It proceeds first by extracting the periodic part using any of the numerousexisting techniques [13,14], then by applying the cyclic regression to enhance the second-order cyclostationarypart. The so-called SUBLEX method described in [15] was designed to substitute for the BSS methods incomplex mechanical systems using the convolutive mixture model. It estimates the array response vectorcorresponding to the SOI by computing the cyclic spectrum of the observed signals at the cyclic frequency ofthe SOI and then constructs a subspace projector to achieve the separation. Interestingly, the methodsproposed in [11] and [15] achieve the same objectives under the same assumptions, but from apparentlydifferent approaches.

The purpose of this paper is to propose a unifying approach on the basis of the reduced-rank cyclicregression (RRCR) [16]. The motivation of this paper is to design a method that benefits from the respectiveadvantages of the original MCR and SUBLEX methods but suppresses their drawbacks. Just like MCR andSUBLEX, RRCR requires the knowledge of the cyclic frequency of the SOI to be extracted, which can bemeasured from the observed signals, for instance by means of the spectral correlation or the squared envelopespectrum. Hence, it is blind in the sense that nothing else is assumed neither on the interferences nor on thenoise.

The paper is organised as follows. In Section 2, a review of the BSS issue, its formulation and its limitationsis presented. In Sections 3 and 4, a description of recently developed methods with their advantages anddrawbacks is presented. Section 5 proposes a unifying approach based on RRCR. In Section 6, itsperformance is assessed by means of numerical simulations. Finally, the last section is dedicated to theapplication of the method to real industrial signals for bearing and gearbox diagnostic purposes.

2. Problem statement

2.1. The BSS issue

In the following, all signals are assumed of finite-power, zero-mean1 and stochastic. BSS consists ofrecovering m unknown source signals, sðtÞ ¼ ðs1ðtÞ; s2ðtÞ; . . . ; smðtÞÞ

T solely from the knowledge of n observation

signals xðtÞ ¼ ðx1ðtÞ;x2ðtÞ; . . . ; xnðtÞÞT , t 2 R. The convolutive mixture model, which describes well transfer

paths in mechanical structures will be adopted hereafter such that

xiðtÞ ¼Xm

j¼1

hijðtÞ � sjðtÞ þ niðtÞ; i ¼ 1; . . . ; n (1)

or in a matrix form

xðtÞ ¼ HðtÞ � sðtÞ þ nðtÞ, (2)

where � is the convolution product, HðtÞ denotes a full-rank matrix of time-invariant linear filters hijðtÞ

modelling the transfers from sources i to sensors j, and nðtÞ is a vector containing additive noise signals

1This assumption is for simplicity, but it is not a fundamental requirement.

ARTICLE IN PRESSR. Boustany, J. Antoni / Mechanical Systems and Signal Processing 22 (2008) 520–541522

mutually uncorrelated and uncorrelated with the source signals sðtÞ. The model described above may also bewritten in the frequency domain as

dXðf Þ ¼ Hðf ÞdSðf Þ þ dNðf Þ, (3)

where dXðf Þ (resp. dSðf Þ and dNðf Þ) is a vector containing the so-called spectral increments fdX iðf Þgni¼1 (resp.

fdSjðf Þgmj¼1 and fdNiðf Þg

ni¼1) of Cramer’s decomposition2 of the stochastic signals fxiðtÞg

ni¼1 (resp. fsjðtÞg

mj¼1 and

fniðtÞgni¼1). Classical BSS assumes the knowledge of the number of source signals present in the mixture and

aims at recovering all of them, usually under the sole assumption of their mutual statistical independence.However, in many situations, the number m of sources is unknown and their statistical independence may notbe respected, and yet there is only one signal of interest (SOI) to be recovered. For instance, this is typically thecase in vibration-based condition monitoring plants, where the SOI is the fault signal stemming from oneparticular mechanical component among many others. This is also typically the case in noise analyses whereone focuses on the noise radiated by one particular source among many other interfering sources. In suchcases, the BSS problem actually reduces to a BSE problem, since the restoration of only one particular sourceis of concern.

This paper deals with the special but important case where the SOI exhibits cyclostationarity.Cyclostationarity is a statistical property that characterises signals which are intrinsically produced bysome periodic phenomena, such as commonly encountered in the natural environment or in industrialapplications. In particular, cyclostationarity has been found to describe very well the nature of acoustical andvibration signals generated by rotating and reciprocating machines [10,17] and therefore has importantpractical implications on improving current signal processing techniques dedicated to vibration and noiseanalyses [11,15,18]. Before proceeding further, a brief review of cyclostationarity and its descriptors ispresented.

2.2. Cyclostationary signals and their second-order descriptors

Cyclostationarity is a sub-class of non-stationarity. A second-order (quasi-) cyclostationary signal sðtÞ withcyclic frequencies ak 2A is such that its second-order statistics are (quasi-) periodic, i.e. they accept a Fourierseries expansion over the set of frequencies in A. For instance, the instantaneous auto-spectral densityS2sðt; f Þ—defined as the Fourier transform of the instantaneous auto-correlation function R2sðt; tÞ ¼EfsðtÞs�ðt� tÞg [19]—expands as [20]

S2sðt; f Þ ¼Xak2A

Sak

2s ðf Þej2pakt, (4)

where the so-called cyclic power spectra Sak

2s ðf Þ are non-identically zero over the set A and are related to thepreviously defined spectral increments as

Sak

2s ðf Þ df ¼ EfdSðf Þ dSðf � akÞ�g. (5)

In words, cyclostationarity extends the usual description of stationary signal [19] to a certain type of non-stationary signals, namely those with a (quasi-) periodic non-stationary structure. This is exactly themeaning of Eq. (4), which distributes the signal power as a function of frequency f and of time t, yetwith a cyclical behaviour embodied by the trigonometric functions ej2pakt with periods 1=ak. A very interestingconsequence of this hidden periodicity is that spectral components are forced to occur in synchronisationso that, contrary to stationary signals,3 they exhibit correlation between adjacent frequencies as materia-lised by Eq. (5). The extra quantity of information provided by cyclostationarity makes it possible to designmore advanced processing techniques than is allowed by the classical but restrictive assumption ofstationarity.

2xðtÞ ¼Rþ1�1

ej2pft dX ðf Þ, where dX ðf Þ may be interpreted as the stochastic Fourier coefficient of xðtÞ at frequency f .3It is well known that for stationary signals EfdSðf 1Þ dSðf 2Þ

�g ¼ S2sðf 1Þdðf 1 � f 2Þ df 1 df 2.

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2.3. Statement of the BSE issue

As explained above, the BSE task arises in many mechanical applications concerned with vibrationor noise analyses, and in such situations there are many instances where the SOI may be reasonablyassumed cyclostationary. From now on, the SOI will be assumed to have cyclic frequencies ak 2Aand, without loss of generality, will be denoted by s1ðtÞ—i.e. the first element in vector sðtÞ. Let us alsodefine:

�

4

mo

the spatially coherent ‘‘undesired’’ signals given by contributionsPm

j¼2Hijðf Þ dSjðf Þ for i ¼ 1; . . . ; n, andwhich will be referred to as ‘‘interferences’’, and

� the spatially non-coherent ‘‘undesired’’ signals given by contributions dNiðf Þ for i ¼ 1; . . . ; n, and which will

be referred to simply as ‘‘noise’’.

The main assumption required to extract s1ðtÞ from the interferences and the noise is that A containsat least one cyclic frequency ak that is not shared with any of the other ‘‘undesired’’ sources fsjðtÞg

mj¼2—be they

stationary or cyclostationary.The following two sections briefly review how the cyclostationarity of the SOI may be exploited in order to

extract it from the other interfering signals [11,15].

3. The SUBLEX method

3.1. Principles of SUBLEX

In a recent work [15], the authors proposed a so-called SUBLEX method for the blind extractionof the SOI by exploiting its cyclostationarity through a subspace decomposition of the data. Contraryto other BSE approaches, the main advantage of SUBLEX is that it does not require any a priori

knowledge on the interferences nor on their number. In this respect, it is a totally blind algorithm.Under the assumptions of Section 2.3, the idea of SUBLEX consists of designing a projection matrix Pðf Þ

such that

dXj1ðf Þ ¼ Pðf ÞdXðf Þ (6)

is an n-dimensional vector as ‘‘close’’ as possible to the contributions dXj1ðf Þ of the SOI on the sensors.4

In [15], it was shown that the projector Pðf Þ can be obtained from the structure Pðf Þ ¼ I� Cðf ÞBðf ÞH whereBðf Þ is an n� ðn� 1Þ unitary matrix that is orthogonal to the eigenvector uðf Þ associated with the greatestsingular value of the cyclic spectral matrix Sa

2xðf Þ of the observations, and where

Cðf Þ ¼ Sxzðf ÞS2zðf Þ�1, (7)

with

dZðf Þ ¼ Bðf ÞHdXðf Þ,

Sxzðf Þ df ¼ EfdXðf ÞdZðf ÞHg,

S2zðf Þ df ¼ EfdZðf ÞdZðf ÞHg.

One suggested improvement in [15] was to compute the projector Pðf Þ on the pre-whitened data, in which caseit could be simply written as

Pðf Þ ¼ S1=22x ðf Þuðf Þuðf Þ

HS�1=22x ðf Þ, (8)

The contribution of a source sjðtÞ on a sensor is the signal as it would be recorded on this sensor if only source sjðtÞ was active, i.e. using

del (1), xijjðtÞ ¼ hijðtÞ � sjðtÞ is the contribution of source j on sensor i.

ARTICLE IN PRESSR. Boustany, J. Antoni / Mechanical Systems and Signal Processing 22 (2008) 520–541524

where uðf Þ is now the eigenvector associated with the greatest eigenvalue of

S�1=22x ðf ÞS

a2xðf ÞS

�12x ðf � aÞSa

2xðf ÞHS�1=22x ðf Þ. (9)

The above theoretical developments suggest two remarks:

Remark 1. Pðf Þ is a rank-one (non-orthogonal) projector, i.e. such that P2ðf Þ ¼ Pðf Þ and rankfPðf Þg ¼ 1.

Remark 2. The SUBLEX method presented so far takes advantage of only one cyclic frequency a.In order to make it more robust, the authors proposed in [15] the joint diagonalisation of severalmatrices of the form (9) indexed by different cyclic frequencies ak 2 B �A. The condition is naturallythat these cyclic frequencies are not shared with any interference or noise signals. Another empiricalway to do so is via (8), with uðf Þ replaced by the eigenvector associated with the greatest eigenvalue of thematrix

S�1=22x ðf ÞSxrðf ÞS

�12r ðf ÞSrxðf Þ

HS�1=22x ðf Þ, (10)

where

dR ¼ ðdXðf � a1ÞT ; . . . ; dXðf � akÞ

T ; . . . ; dXðf � aK ÞTÞT (11)

is an extended observation vector containing frequency-shifted versions of the observation vector dXðf Þ, andK ¼ CardðBÞ.

A formal justification to this solution will be given in Section 5.

3.2. Advantages and drawbacks of SUBLEX

3.2.1. Advantages of SUBLEX

(1)

5k

In accordance with the assumptions of Section 2.3, SUBLEX performs well with the use of only one cyclicfrequency, in the sense that the choice of only one cyclic frequency a 2A is sufficient to extract the whole

signal contributions xj1ðtÞ. The consideration of more than one cyclic frequency as advocated in Remark 2is surely an advantage, but not a necessity.

3.2.2. Drawbacks of SUBLEX

(1)

SUBLEX is a multiple-sensor method. Like all subspace methods, an exact restoration of the SOI requireshaving at least as many sensors as interference signals—i.e nXm—and no noise signals. In all other cases,the restored signal is an approximation of the SOI.

(2)

SUBLEX is explicitly a projection-based method: for all f , the observation data dXðf Þ are projected ontothe signal subspace spanned by dXj1ðf Þ, even in the absence of signals. Therefore in the noisy case,Pðf ÞdNðf Þ may then be significantly different from zero (kPðf Þk40 for any f )5 in regions where the signal-to-noise ratio (SNR) is low, thus likely to provide ‘‘noisy’’ results.

4. The multiple cyclic regression (MCR) technique

4.1. Principles of MCR

In a recent work [11], Bonnardot et al. proposed a tool for denoising and extracting the cyclostationary partof a signal by exploiting its spectral redundancy. The method applies on exactly the same assumptions as thoselisted in Section 2.3, but contrary to SUBLEX, operates in the time domain and proceeds the reconstruction ofthe SOI by combining several frequency-shifted versions of the observation signal. This section presents afrequency-domain version of [11] based on model (3). In the general multiple-sensor case, it aims at finding

:k denotes the Frobenius norm of a matrix.

ARTICLE IN PRESSR. Boustany, J. Antoni / Mechanical Systems and Signal Processing 22 (2008) 520–541 525

a transfer matrix Gðf Þ such that

dXj1ðf Þ ¼ Gðf ÞdRðf Þ, (12)

with

Gðf Þ ¼ ArgminEfkdXðf Þ �Gðf ÞdRðf Þk2g, (13)

where dRðf Þ is the extended observation vector given by (11) and defined in remark 2 over the set of cyclicfrequencies ak 2 B � A. The solution of this regression problem is a cyclic Wiener filter [12] whose expressionreads

Gðf Þ ¼ Sxrðf ÞS�12r ðf Þ. (14)

Two main remarks arise at this stage.

Remark 3. In contrast to SUBLEX, the designed transfer matrix Gðf Þ resulting from MCR is not a projector,i.e. G2

ðf ÞcGðf Þ.

Remark 4. In contrast to SUBLEX, the designed transfer matrix Gðf Þ is not constrained to be of rank 1, i.e.rankfGðf Þga1 in general.

Indeed, the MCR method is in essence a multivariate linear regression method, and this is fundamentallydifferent from a nonlinear subspace method such as SUBLEX. Despite these differences, MCR just as SUBLEXcan achieve perfect extraction of the SOI in the noise-free case, as stated by the following proposition:

Proposition 1. The error of MCR is nil in the noise-free case provided that there exists a finite set of coefficients

fGkðf ÞgKk¼1 at every f -frequency such that

XK

k¼1aka0

Gkðf ÞdXðf � akÞ ¼ dXðf Þ; ak 2 B �A. (15)

It now remains to analyse the advantages and drawbacks of MCR in the most general case, and to comparethem to those of SUBLEX.

4.2. Advantages and drawbacks of MCR

4.2.1. Advantages of MCR

(1)

6T

MCR can cope with single-sensor as well as multiple-sensor problems, contrary to SUBLEX which makesuse of at least two sensors, and more generally requires at least as many sensors as sources for perfectreconstruction.

(2)

By construction, the transfer function of MCR is such that Gðf Þ � 0 in regions where the SNR is very low.This implies that MCR denoises the signal better than SUBLEX in the presence of noise signals.

4.2.2. Drawbacks of MCR

(1)

Experimental results show that the spectral matrix of the recovered SOI is in general not of unitary rank asit should theoretically be,6 i.e. rankfSxj1ðf ÞgX1 as opposed to rankfSxj1 ðf Þg ¼ 1. As a consequence thealgorithm in its raw version may lose some statistical efficiency, as will be demonstrated below.

(2)

The choice of the optimal number of cyclic frequencies to be used in Eq. (14) is not straightforward,since it depends on a compromise between two conflicting requirements. The first requirement is thata sufficient number of cyclic frequencies should be used—at least K—so that Proposition 1 is satisfied.The second requirement is that as few cyclic frequencies as possible should be retained in order to improvethe filter estimation: the more parameters to estimate, the higher the estimation variance. This last

he rank is theoretically equal to the number of SOIs, that is 1 in the present work.

ARTICLE IN PRESSR. Boustany, J. Antoni / Mechanical Systems and Signal Processing 22 (2008) 520–541526

requirement is given a more precise statement through the following proposition, whose proof is providedin Appendix A.

Proposition 2. Let Gðf Þ be the estimated MCR transfer function (Eq. (14)) on signals of length L. Let us also

define the normalised mean-square error NMSE as the ratio of the powers after and before application of Gðf Þ.Then, in the absence of signal,

NMSESNR¼0 ¼EfkGðf ÞdRðf Þk2g

EfkdRðf Þk2g/

nKa

L, (16)

where n stands for the number of sensors, and Ka for the number of cyclic frequencies.

The latter proposition clearly establishes that

�

the noise rejection capability of MCR is only asymptotic, and may significantly degrade with short-lengthsignals, � the noise rejection capability of MCR degrades as the number n of sensors is increased, � the noise rejection capability of MCR degrades as the number Ka of cyclic frequencies is increased.

These important remarks will be extensively illustrated by numerical simulations in Section 6. They are alsothe core of our motivation to conceive the improved algorithm presented in the next section, since the reasonsof these deficiencies will be traced back to Sxj1 ðf Þ not being of rank 1.

5. A unified approach based on reduced-rank cyclic regression (RRCR)

5.1. Principles of RRCR

The previous sections dealt with the presentation of SUBLEX and MCR, as well as their advantages anddrawbacks. They showed that the two recently developed methods are dedicated to the same objective, namelythe extraction of a cyclostationary signal corrupted by an unknown number of undesired interferences. Theyalso demonstrated that the advantages of one method are actually the drawbacks of the other one. Thequestions that naturally arise at this stage are the following:

(1)

Is it possible to combine in some way SUBLEX and MCR so as to benefit from their respective advantagesand suppress their drawbacks?

(2)

If yes, what would then be the optimal combination?

More specifically, such an optimal combination would have to benefit from the listed advantages of MCR, butwould also have to guarantee that rankfSxj1 ðf Þg ¼ 1—alike SUBLEX. As will be recognised later, thisrequirement will actually turn out as an efficient solution to stabilise the statistical behaviour of the estimatesindependently of the number of sensors and cyclic frequencies. Therefore our objective is to find a transfermatrix Wðf Þ such that the estimation

dXj1ðf Þ ¼Wðf ÞdRðf Þ (17)

is as ‘‘close’’ as possible to the actual contributions dXj1ðf Þ (alikeMCR) under the constraint of producing a spectraldensity matrix Sxj1 ðf Þ of rank 1 (alike SUBLEX). This problem can be tackled by searching for the solution of

Wðf Þ ¼ ArgminEfkC�1ðdXj1ðf Þ �Wðf ÞdRðf ÞÞk2g, (18)

under the equivalent constraint that rankfWðf Þg ¼ 1, with C a suitably chosen positive-definite hermitian matrix

[16]. A classical choice is to set C ¼ S1=22x ðf Þ in Eq. (18), which achieves the minimum variance of Wðf Þ. This gives

the two equivalent solutions [16]:

Wðf Þ ¼ S1=22x ðf Þuðf Þuðf Þ

HS�1=22x ðf ÞSxrðf ÞS

�12r ðf Þ (19)

ARTICLE IN PRESSR. Boustany, J. Antoni / Mechanical Systems and Signal Processing 22 (2008) 520–541 527

or

Wðf Þ ¼ Sxrðf ÞS�1=22r ðf Þvðf Þvðf Þ

HS�1=22r ðf Þ, (20)

where uðf Þ and vðf Þ are the eigenvectors associated with the greatest eigenvalues of

S�1=22x ðf ÞSxrðf ÞS

�12r ðf ÞSrxðf ÞS

�1=22x ðf Þ (21)

and

S�1=22r ðf ÞSrxðf ÞS

�12x ðf ÞSxrðf ÞS

�1=22r ðf Þ,

respectively.

5.2. Interpretation of RRCR

From Eq. (19), it is clear that the first optimal solution combines the solutions of SUBLEX and MCR suchthat Wðf Þ ¼ Pðf ÞGðf Þ. Hence, the interpretation of RRCR is that of a serial combination of the transferfunction of MCR with the projection matrix of SUBLEX, as illustrated in Fig. 1. This provides one possibleoptimal combined structure. From the second equation (20), another optimal solution is Wðf Þ ¼ Gðf ÞQðf Þ,where Gðf Þ is again the MCR transfer function and Qðf Þ ¼ S

1=22r ðf Þvðf Þvðf Þ

HS�1=22r ðf Þ is a projector equivalent

to that of SUBLEX. Let us emphasize again that these two optimal solutions are equivalent in the sense thatthey achieve exactly the same minimum in Eq. (18).

Remark 5. In the single-sensor case, the optimal solution (19) reduces to MCR.

Remark 6. The structure of the optimal solution (19) now formally justifies the form of Eq. (10) in Remark 2.

The interpretation of RRCR as the cascade of MCR and SUBLEX suggests that it will combine therespective advantages of the two latter methods. Surprisingly, it also fixes their respective drawbacks. This isjustified by the following proposition—see proof in Appendix B—which establishes that RRCR significantlyoutperforms the behaviour of both SUBLEX and MCR in regions where the SNR is low.

Proposition 3. Let Wðf Þ be the estimated RRCR transfer function (Eq. (19) or (20)) on signals of length L. Then,in the absence of signal, the NMSE is

NMSESNR¼0 ¼EfkWðf ÞdRðf Þk2g

EfkdRðf Þk2g/ 1=L, (22)

that is, it tends to zero as 1=L independently of the number of sensors and cyclic frequencies.

Therefore, in addition to the fact that RRCR provides a unifying approach to such different approaches asMCR and SUBLEX, it appears that it also achieves a better extraction of the SOI.

A more intuitive interpretation of RRCR is presented in the following summary and illustrated in Fig. 1.In order to recover the SOI, RRCR proceeds as follows:

(1)

RRCR uses the observed signals dXðf Þ and only requires measuring of at least one cyclic frequency a1 ofthe SOI dXj1ðf Þ. As indicated before, this may be performed by computing the spectral correlation of oneelement of vector dXðf Þ.

MCRSUBLEX

Cyclic

Wiener

filter G(f)

Frequency

shifting

Rank-one

projector

P(f)dX(f) dR(f) dX

|1(f)

W(f)

Measurement

of αk

Fig. 1. Block diagram of the combined regression and projection method.

ARTICLE IN PRESSR. Boustany, J. Antoni / Mechanical Systems and Signal Processing 22 (2008) 520–541528

(2)

The extended observation vector dRðf Þ is then constructed using frequency-shifted versions of dXðf Þ asgiven by Eq. (11).

(3)

The cyclic Wiener filter is first applied to dRðf Þ yielding a first estimation of dXj1ðf Þ. It operatesby extracting the signal component that is correlated in reference vector dRðf Þ. Therefore, sincethe noise dNðf Þ is neither correlated in frequency nor spatially coherent, it is not extracted from dRðf Þ.Hence, the filter Gðf Þ is destructive to the additive noise, thus improving significantly the SNR.Unfortunately, it does not provide satisfactory interference rejection. This is due to the fact that althoughthe frequency-shifted versions of the interferences are not correlated, still the interferences are spatiallycoherent.

(4)

Adding the rank-one subspace projector Pðf Þ remedies to this problem since it insures that only thecontribution of the cyclostationary SOI is recovered at its output and thus increases considerably thesignal-to-interference ratio.

The previous statements and remarks will be verified through extensive simulations in the next section.

6. Performance assessment through simulations

The aim of this section is to assess the performance of RRCR by means of simulations, and to compare itwith the performances of SUBLEX and MCR. In order to do so, a mixture of sources in presence of additivenoise is modelled according to Eq. (1). The synthesised SOI is cyclostationary with a cyclic period denoted byT, and its discrete version s1½n� of length L is modelled as

s1½n� ¼XbL=Tc

l¼1

Ald½n� lT � tl �, (23)

where d½n� is the discrete Dirac impulse, Al is a random amplitude following the normal law NðmA;s2AÞ, tl is a

sequence of independent and identically distributed (iid) random variables following the normal law Nð0;s2tÞ.Next, the interfering source signals fsj ½n�g

mj¼2 and the additive noise signals fni½n�g

ni¼1 were synthesised from

random sequences distributed like Nð0;s2j Þ and Nð0;s2nÞ, respectively. Finally, the linear filters fhji½n�g are

modelled by impulse responses of second-order systems. Note that signal (23) models very well a number ofrotating machine vibrations, especially in the case of incipient faults [10].

The performance of the three methods described in this paper are addressed with respect to five parametersof interest:

(1)

the number of frequency shifts (or cyclic frequencies) Ka, (2) the number of sensors n,

2

(3)

the noise-to-signal ratio NSR ¼ 10 logðsn

s21

Þ, Pms2

(4)

the interference-to-signal ratio ISR ¼ 10 logð j¼2 j

s21

Þ,

(5) the standard deviation st of the iid sequence tl .

6.1. Evaluation of the error versus the number of cyclic frequencies

The first experiment consists of investigating the behaviour of the three methods with respect to the numberof cyclic frequencies (or frequency shifts) Ka used. The parameter settings for this experiment are thefollowing:

�

number of sources m ¼ 4 including the SOI, � n ¼ 4, � mA ¼ 1 and sA ¼ 0:1, � st ¼ 0, � L ¼ 16 384 and T ¼ 256 samples,

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Fig. 2. Normalised mean square error versus the number of frequency shifts Ka.

R. Boustany, J. Antoni / Mechanical Systems and Signal Processing 22 (2008) 520–541 529

�

ISR ¼ �10 dB, � NSR ¼ �20 dB, � Ka 2 B ¼ ½1; 10�.

Results are displayed in Fig. 2. They reveal a better performance of the RRCR method compared to SUBLEXand MCR. Most importantly, the error curve of MCR is found to increase linearly with respect to Ka, thusconfirming Proposition 2, whereas that of error of RRCR is relatively constant, thus confirming Proposition 3.Note also that RRCR clearly achieves the smallest NMSE, with a significant advantage (factor 6) overSUBLEX.

Incidentally, Ref. [11] also reported performance curves of MCR versus the number of cyclic frequencies,but with a decreasing tendency contrary to a linear increase as demonstrated here. This is because thesimulations in [11] consider a different model of the mixture and were carried out on a signal with a high SNRand thus adding cyclic frequencies decreased the NMSE till a certain point according to Proposition 1. Butafter that point—which is not shown in [11]—NMSE increase according to Proposition 2. On contrary, oursimulations are performed with a low SNR where the effects of 2 predominates.

6.2. Evaluation of the error versus the number of sensors

Next, the three methods are compared with respect to the number of sensors used for the extraction of theSOI. All the parameters of the previous experiment are kept unchanged, but the number of frequency shifts isnow fixed to Ka ¼ 2 and the number n of sensors is allowed to vary in the range ½1; 10�—note that SUBLEXneeds at least 2 sensors to work. By examining the results in Fig. 3, it is obvious that RRCR is also very robustwith respect to the number of sensors, whereas the error of MCR increases almost linearly with n—here againas predicted by Proposition 2. It remains to mention that when n ¼ 1 RRCR still works—it is actuallyequivalent to MCR—but its performance is not better than when it is combined to SUBLEX as it may be seenwhen nX2.

6.3. Evaluation of the error versus the noise-to-signal ratio NSR

The aim of this third experiment is to assess the performance with respect to the noise-to-signal ratio. TheNSR is allowed to take values in the interval ½�20;�5�dB, all the other parameters being kept constant and in

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Fig. 3. Normalised mean square error versus the number of sensors n.

Fig. 4. Normalised mean square error versus the noise-to-signal ratio NSR.

R. Boustany, J. Antoni / Mechanical Systems and Signal Processing 22 (2008) 520–541530

particular n ¼ 4 and Ka ¼ 2. Simulation results are depicted in Fig. 4 and show that SUBLEX is less robust tonoisy mixtures than MCR, but that RRCR still works well even for high NSR values. This is again inaccordance with the theoretical analyses of Sections 3–5.

6.4. Evaluation of the error versus the interference-to-signal ratio ISR

Another parameter of interest is the ISR. In this simulation, ISR took values in the interval ½�12; 0�dB,whereas NSR was fixed to �20 dB. Results are presented in Fig. 5. In contrast to the previous results, MCR isnow less robust to interferences than SUBLEX—SUBLEX is specifically constructed to take advantage ofthese interferences—but RRCR still shows the best behaviour.

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Fig. 5. Normalised mean square error versus the interference-to-signal ratio ISR.

Fig. 6. Normalised mean square error versus the percentage of cyclic period fluctuation st=T .

R. Boustany, J. Antoni / Mechanical Systems and Signal Processing 22 (2008) 520–541 531

6.5. Evaluation of the error versus the variability of the cyclic period

In some contexts, the cyclic period may be constant only to some degree of accuracy. As indicated byEq. (23), the robustness of the methods with respect to the accuracy of the cyclic frequency may be studiedthrough the standard deviation st of variables tl . The result of this analysis are reported in Fig. 6, which showsthat RRCR has again the best performance, and that SUBLEX is more robust to uncertainty in the cyclicperiod than MCR. Further research is in progress to assess these interesting behaviours from a theoreticalpoint of view.

7. Application to industrial data

The previous sections have clearly confirmed through both theoretical and simulation results the superiorperformance of RRCR for extracting an unknown cyclostationary SOI from interfering sources and additive

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noise. It now remains to illustrate the proposed methodology on actual industrial signals. The followingexamples are concerned with applications to rolling-element bearing and gearbox diagnostics. It is worthinsisting on the fact that several well-accepted methods exist to solve these problems—e.g. spectral, cepstral,envelope analyses—and that the principle of blind signal extraction is not to concurrence them, but rather tocomplement them either as a pre-processing tool for signal enhancement and denoising, or as a post-processing tool for better visual inspection of the incriminated signal.

7.1. Rolling-element bearing diagnostics

In this first application example, we consider measurements taken on a one-stage gearbox [22], with a ratioof 32 : 49, an input shaft speed of f r ¼ 3Hz, and a torque of 60Nm. The driving shaft is supported by adouble-row self-aligning ball bearing whose characteristics are the following:

�

Fig

fau

f -f

outer race diameter ¼ 44:85mm;

� inner race diameter ¼ 32:17mm; � ball diameter Bd ¼ 7:12mm; � number of balls Nb ¼ 12=row.

The data set consists of 7 sensor records of 100 000 points each with a sampling frequency f s ¼ 16 384Hz. Theobjective is to extract the contribution (time-waveform) of a fault caused by a slot on the inner race of the ballbearing. To do so, we first need to set the value of the cyclic frequency a of the SOI. Indeed, the theoreticalvalue of the ball-pass frequency on a localised defect on the inner race (BPFI) suggests thata ¼ BPFI � 1:36� 10�3f s � 22:3Hz. However, this value is subjected to measurement errors on the shaftspeed, the contact angle, and the omission of possible slip of the cage. A more accurate procedure to evaluatethe cyclic frequency is to inspect the estimated spectral correlation of an observation signal [15]. Fig. 7 showshigh spectral correlation values at cyclic frequency ag ¼ 6:12� 10�3f s corresponding to the gear meshfrequency and at a ¼ BPFI ¼ 1:36� 10�3f s � 22:28Hz linked to the bearing fault. Again, we point out to the

. 7. Spectral correlation of a sensor signal computed with a half-sine window of 256 samples and 75% overlap showing a ball bearing

lt cyclic frequency at a ¼ 1:36� 10�3f s. The signal is pre-whitened and only values above a 5% significance level are displayed. The

requency resolution is Df ’ 4:80� 10�3f s and the a-frequency resolution is Da ’ 1:00� 10�5f s.

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reader that the so-computed spectral correlation is in principle enough to diagnose the system [21], but it is notenough to return the time-waveform of the fault. It is also a mean of measuring the cyclic frequency of the SOIprior to the application of the proposed extracting techniques.

Fig. 8 displays the eigenvalues of matrix (21) of the observations. It is well known that there is theoreticallyonly one non-zero eigenvalue corresponding to the cyclostationary SOI. Practically, many eigenvalues mayhave significant amplitudes. This is due to at least two reasons: mainly to the estimation errors resulting fromfinite length effects and spectral leakage, and to natural fluctuation of the cyclic frequency. However, it can beseen that there is still one eigenvalue that is dominating over all the others. Figs. 7 and 8 also indicate very wellthe frequency band of the high-pass filter to apply on the measured signals prior to the application of themethods. A pre-filtering in the frequency band ½0:25; 0:5�f s is sufficient to enhance the SNR (just as in [15]prior to application of SUBLEX). Fig. 9 shows, respectively, one of the high-pass filtered sensor signals andthe corresponding extracted SOI using SUBLEX, MCR and RRCR. It is obvious that SUBLEX (b) andMCR (c) both perform well—as reported in Refs. [11] and [15], respectively, and actually return quite similarresults. Nevertheless careful inspection of the time waveforms shows that the RRCR solution (d) enjoys asuperior degree of accuracy, in the sense that it best enhances the impacts due to the slot on the inner race ofthe ball bearing. Fig. 10 displays a zoom on 4 impacts (4 cycles) characterising the fault on the inner. Similarlyto what was presented in [15], one can also verify through a squared-envelope analysis that the latter impactsare modulated by the load at the rotation frequency.

7.2. Gearbox diagnostics

The second application illustrates the use of the RRCR method as a tool for the diagnostics of a two single-stage gearbox units mounted back to back. This subject was already addressed in [23], where it was shown thatclassical synchronous averaging techniques can be used as a first-order cyclostationary (CS1) study of thevibration signal but they neglect the second-order cyclostationary (CS2) information contained in it. Theauthors of [23] proposed to use the spectral correlation to characterise the lost CS2 component. However thesampling frequency of interest was limited to a region where the CS1 part was dominant. In a recent work [24],the authors extended the study to a larger frequency band to characterise second and higher-order cyclicstatistics of the fault. Indeed, the magnitude spectrum of such a signal (Fig. 11) reveals the presence of two

Fig. 8. Eigenvalues of matrix (21). The f -frequency resolution is Df ’ 1:9� 10�2f s.

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Fig. 9. Extraction of the localised bearing defect. (a) Sensor signal. (b) Bearing fault signature from SUBLEX. (c) Bearing fault signature

from MCR. (d) Bearing fault signature from RRCR.

R. Boustany, J. Antoni / Mechanical Systems and Signal Processing 22 (2008) 520–541534

clearly separated regions. The first region (Fig. 11.b and up to 10 kHz) is dominated by CS1 components andmay be linked to ‘‘macroscopical’’ phenomena or large-scale deformations in the mechanical system—i.e.vibrations due to the rotating shaft and those due to the meshing of the gears. The second region in a bandaround 23 kHz (Fig. 11.c) is mainly composed of CS2 random components and may be related to‘‘microscopical’’ or local phenomena or small-scale deformations in the machine parts—i.e. vibrations due tofriction and contact between tooth surfaces. Hence, the evolution of small-scale defects like pittings to spallwill be more traceable in the high-frequency CS2 region and may lead to an early diagnosis of the gears. Noteagain that the spectral correlation is a sufficient tool to diagnose the mechanical system as advocated in [21,23]and that the present experiment aims at extracting the time-waveform of the fault signature which the spectralcorrelation obviously cannot do.

The data set consists of two vibration signals taken on a system similar to that of [23] with a gear ratio20:20, and of a phase signal for order tracking. All the signals are sampled at f s ¼ 80 kHz and are of lengthL ¼ 160 000 samples each. In order to eliminate the randomness induced by the fluctuation of the rotatingfrequency, the signals were first angularly re-sampled. The angular re-sampling technique, its advantages andlimitations when applied to rotating machine applications is discussed in Refs. [10,18,25].

Fig. 13 displays the eigenvalues of matrix (21) of the observations. Just as in previous example of Section7.1, it can be seen that there is one eigenvalue that is dominating over the other one. Figs. 12 and 13 allow thechoice of the frequency pass-band of the filter to apply on the measured signals prior to the application of

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Fig. 10. Few cycles of (a) one sensor signal, the bearing fault signature from (b) SUBLEX, (c) MCR and (d) RRCR revealing the typical

impacts modulated by the load and characterising the fault on the inner race of the bearing.

R. Boustany, J. Antoni / Mechanical Systems and Signal Processing 22 (2008) 520–541 535

RRCR. A good choice is to consider the frequency band ½20; 40�kHz.7 Next, the spectral correlation of onevibration signal was computed and is illustrated in Fig. 12. It shows high levels of cyclostationarity at therotation frequency a ’ 16:1Hz linked to the spall on one tooth of the gear. This cyclic frequency is now used

7Contrary to the previous example of Section 7.1, Fig. 13 alone is not sufficient for the choice of the frequency pass-band of the filter,

because in the present case, the gear mesh frequency is a multiple of the cyclic frequency of the SOI at which matrix 21 was computed, and

thus it also appears in low-frequency regions around 0:18f s.

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16 18 20 22 24 26 28 300 0.5 1 1.5 2 2.5 3 3.5

0 5 10 15 20 25 30 35 400

Frequency [kHz]

Am

plitu

de (

m/s

2/H

z)Magnitude spectrum of the vibration signal

Frequency [kHz]Frequency [kHz]

a

b c

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0

1

2

3

4

5

6

7

8

9x 10-3

Fig. 11. Magnitude spectrum of a vibration signal from a gearbox test bench.

R. Boustany, J. Antoni / Mechanical Systems and Signal Processing 22 (2008) 520–541536

to extract the fault signature using RRCR method. The rows of Fig. 14 show respectively the sensor signalsand the corresponding extracted SOI using RRCR after pre-filtering. The time-waveform of the extractedsignal exhibits clearly the presence of the spall on the accelerometer signal. It also informs the presence ofmore than one spalled tooth: an additional information which is not traceable in a spectral correlationanalysis. Indeed, the Wigner–Ville spectrum [18] of one of the extracted signals was computed and is displayedin Fig. 15. The temporal information indicates the location of the developed defects on the 1st, 5th and 6thteeth. The frequency information informs us that the frequency band concerned by the fault is ½20; 30�kHz asexpected by inspection of the spectral correlation (Fig. 12).

8. Conclusion

In two precursory works, a subspace and a cyclic regression methods for the blind extraction of a second-order cyclostationary signal were developed. The present paper proposed a unifying approach based on

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Fig. 12. Spectral correlation of a sensor signal computed with a half-sine window of 256 samples and 75% overlap showing a gear fault

cyclic frequency at a ’ 16:1Hz. The signal is pre-whitened and only values above a 5% significance level are displayed. The f -frequency

resolution is Df ’ 0:384kHz and the a-frequency resolution is Da ’ 0:5Hz.

Fig. 13. Eigenvalues of matrix (21). The f -frequency resolution is Df ’ 1:9� 10�2f s.

R. Boustany, J. Antoni / Mechanical Systems and Signal Processing 22 (2008) 520–541 537

reduced-rank cyclic regression which benefits from the respective advantages of the two precursory methodsbut suppresses their drawbacks. Moreover and unlike MCR, the power of the additive noise at the output ofRRCR does not depend neither on the number of frequency shifts used in the regression nor on the number ofmeasured signals. These properties were confirmed by numerical simulations which also compared theproposed technique to the former ones. The method was finally applied to bearing and gearbox diagnosticpurposes where it achieves very good extraction of fault signatures.

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Fig. 14. Extraction of the gear spall signature from two measurements using RRCR. (a) Measured signals and (b) extracted signals.

Fig. 15. Wigner–Ville spectrum of one of the extracted signals. The frequency resolution is Df ¼ 1:2kHz.

R. Boustany, J. Antoni / Mechanical Systems and Signal Processing 22 (2008) 520–541538

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Appendix A. Proof of Proposition 2

For notational simplicity, the dependence on the frequency variable f is omitted whenever possible. Theobjective is to evaluate the quantity

NMSESNR¼0 ¼EfkGdRk2SNR¼0g

EfkdXk2SNR¼0g

¼traceEfGS2rG

Hg

traceEfS2xg.

As mentioned earlier, the solution of the MCR problem is the cyclic Wiener filter G ¼ SxrS�1

2r , where^ð:Þ means

that theoretical quantities have been estimated from finite length measurements, as used in Proposition 2.Under the assumption that noise only is present (SNR ¼ 0), and neglecting the bias due to leakage effects inthe spectral estimators, it is straightforward that

Sxr�!L

Sxr 0;

S2r�!L

S2rc0;

8><>:

(A.1)

where the �!L

stands for the probability limit as the signal length L tends to infinity. Then, the followingapproximation asymptotically holds

G ’ DSxrS�12r , (A.2)

where DSxr is the estimation error on Sxr and may be evaluated using the averaged periodogram (equivalentlythe smoothed periodogram8) estimator as

DSxr ¼ Sxr � Sxr

¼1

P

XP

p¼1

XðpÞw RðpÞHw � 0

¼1

P

XP

p¼1

XðpÞw RðpÞHw ,

where XðpÞw is the short-time Fourier transform of the pth segment weighted by some data-window wðtÞ [26]

EfGS2rGHg ¼ EfDSxrS

�12r DS

Hxrg

¼1

P2

XP

p¼1

XP

l¼1

EfXðpÞw RðpÞHw S�12r RðlÞw XðlÞHw g.

Since the two random variables XðpÞw and RðlÞw are asymptotically gaussian and mutually independent for pal

and for 0% overlap,9 the latter equation becomes

EfGS2rGHg ¼

1

P2

XP

p¼1

Xlap

EfXðpÞw RðpÞHw gS�12r EfRðlÞHw XðlÞHw g þ

1

P2

XP

p¼1

Xl¼p

ðEfXðpÞw XðlÞHw gEfRðpÞHw S�12r R

ðlÞw g

þ EfXðpÞw RðpÞHw gS�12r EfRðpÞw XðpÞHw gÞ,

8The final result does not change in its structure.9Again having non-zero overlap can be shown not to change the structure of the final result, but only its scaling.

ARTICLE IN PRESSR. Boustany, J. Antoni / Mechanical Systems and Signal Processing 22 (2008) 520–541540

then,

EfGS2rGHg ¼

1

P2

XP

p¼1

Xl¼p

EfXðpÞw XðpÞHw gEfRðpÞHw S�12r RðpÞw g

¼1

P2

XP

p¼1

S2xtracefS�12r EfR

ðpÞw RðpÞHw gg

¼1

P

XP

p¼1

S2xtracefInKag

¼nKa

PS2x,

which in turn yields

tracefEfGS2rGHgg ¼

nKa

PtracefS2xg,

/nKa

LtracefS2xg.

Finally, the NMSE for SNR ¼ 0 is found to be

NMSESNR¼0 /nKa

L.

Appendix B. Proof of Proposition 3

The objective is to evaluate quantity

NMSESNR¼0 ¼EfkWdRk2SNR¼0g

EfkdXk2SNR¼0g

¼traceEfWS2rW

Hg

traceEfS2xg,

WS2rWH¼ ðWþ DWÞS2rðWþ DWÞH

¼ DWS2rDWH ,

whereDW ¼ DQGþQDG ¼ QDG,

because G 0 when SNR ¼ 0, then

WS2rWH¼ DGQS2rQ

HDGH .

Using the results of Appendix A

EfDGQS2rQHDGH

g ¼ EfDSxrS�12r QS2rQ

HS�12r DSHxrg

and with Q the projector whose expression is given in Section 5

EfDGQS2rQHDGH

g ¼1

KS2xtracefvv

Hg ¼1

KS2x /

1

LS2x.

Finally,

NMSESNR¼0 /1

L.

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