Anovel methodfortheoptimalbandselectionforvibrationsignaldemodulationandcomparisonwiththeKurtogramTomaszBarszcz,AdamJab"on skiDepartmentofMechanicalEngineeringandRobotics,AGHUniversityofScienceandTechnology,Al. Mickiewicza30,30-059Krakow,Polandarticle infoArticlehistory:Received29 June2009Receivedin revisedform25 May2010Accepted27May2010Availableonline 2June2010Keywords:Rolling bearingdiagnosticsSpectralkurtosisNarrowbandamplitudedemodulationabstractThe narrowband amplitude demodulation of a vibration signal enables the extraction ofcomponentscarryinginformationaboutrotatingmachinefaults. However, thequalityofthedemodulatedsignaldependsonthefrequencybandselectedforthedemodula-tion. The spectral kurtosis (SK) was proved to be a very efcient method for detection ofsuchfaults, includingdefectiverollingelementbearingsandgears[1].Althoughthereare conditions, under which SK yields valid results, there are also cases, when it fails, e.g.in the presence of a relatively strong, non-Gaussian noise containing high peaks or for arelativelyhighrepetition rateoffaultimpulses.Inthis paper, anovel methodfor selectionof theoptimal frequencyband, whichattemptstoovercome theaforementioneddrawbacks, ispresented.Subsequently, anewtool forpresentationof resultsof themethod, calledtheProtrugram, isproposed. Themethod is based on the kurtosis of the envelope spectrum amplitudes of the demodulatedsignal, rather than on the kurtosis of the ltered time signal. The advantage of the methodistheabilitytodetecttransientswithsmallersignal-to-noiseratiocomparingtotheSK-basedFast Kurtogram. The application of the proposed method is validatedon simulatedand real data, including a test rig, a simulated signal, and a jet engine vibration signal.& 2010 Elsevier Ltd. All rights reserved.1. IntroductionManyfaultsofrotatingmachineryparts, includingrollingelementbearings(REB)andgeardefectsexcitefrequencyresonances[2]. Whenmachinefault occurs, theseresonancesareexcitedat aspecicrate, inthecaseof REBcalledbearing characteristic frequencies. Consequently, diagnostics of REB by means of envelope analysis1amounts todetection ofbearings characteristic frequencies.In the process of diagnosis, kurtosis is one of most important means of obtaining signatures for machinery faults [2], asit detects fault-induced peaks (or transients) in vibration signals. Generally, the signal components containing informationabout a bearing fault are of relatively low amplitudes, and it was best to apply kurtosis to the band where the dB spectralchangewasgreatest, but thisrequiredhistorical data. Next, themethodof spectral kurtosis(SK) wasproposed. Thismethod is very successful in the detection of rolling bearing faults and tooth cracks in gears. The rst ideas of application ofthespectralkurtosisweregivenbyDwyer[3,4].Ingeneral,theSKisamethodfordetectionofaseriesofimpulsesinasignal. The method can yield a band in which the signal should be demodulated in order to extract the peaky component.ContentslistsavailableatScienceDirectjournal homepage: www.elsevier.com/locate/jnlabr/ymsspMechanicalSystemsandSignal Processing0888-3270/$ - seefrontmatter&2010ElsevierLtd. Allrightsreserved.doi:10.1016/j.ymssp.2010.05.018Correspondingauthor.E-mailaddresses: tbarszcz@agh.edu.pl,tbarszcz@energocontrol.pl (T.Barszcz).1In diagnostics, the term envelope analysis is used interchangeably with the term high frequency resonance techniques, since they both refer tothe frequencyanalysisof theamplitudedemodulatedsignal.MechanicalSystemsandSignalProcessing25 (2011)431451ThedeeperhistoryofthemethodandthetheoreticalformulationoftheSKandestimationmethodsweregivenbyAntoni in[5].Thebasisoftheapproachwasinterpretation ofavibrationsignalasaconditionally nonstationary processand its decomposition based on the WoldCramer theorem. Another important contribution was the formal proposal of anSTFT based SK estimator. In the following paper Antoni and Randall [5,1] showed the practical ability of SK in detection andidentication offaults, eveninthepresenceof high levelnoise.The proposed STFT estimator was valid under conditions of local stationarity and the relatively short correlation lengthofthesignal incomparison withtheSTFT window. Theestimatorhas theform [5]:KYf fsfdg4wNWKX 32_ _11rf 21where KY(f) is theSKestimator, fsthesamplingrate, fdtherepetitionrateof sought impulses, KXtheintensityofuctuationsinimpulseamplitudes, NWtheSTFTwindowlength, g4wthetimebandwidthproductofthesquareoftheanalysiswindow and r(f)thesignal-to-noiseratio of thesignal.Theassumptionsabout conditionswill bediscussedlater inthepaper. It isimportant tonotethat theestimatordependsonsignalparameters suchasuctuations inimpulseamplitudes (whichvery oftenexistinmeasured signals).Tondtheoptimalband, AntoniandRandallproposedthemethodoftheKurtogram, whichpresentsSKvaluesinavisual form on a 2D plane as a function of the central frequency and the bandwidth of the ltered signal. The Kurtogramhelps in a quick determination of optimal lter parameters for signal demodulation, without a priori knowledge about theobject. Inafollowingnextpaper[6], AntoniproposedanothertooltheFastKurtogram, whichinvestigatedonlyafewselectedbandwidthsto obtain theKurtogram andreducedtherequired CPUload.TheSKwasthenappliedbyotherresearchers.CombetandGelman[7]appliedthemethodforthedetectionoflocalfaults ingears. Their approachcombinedtime synchronous averaging (TSA) for extractionof dominating meshingcomponents and analysis of the residual components. The analysis would be the demodulation of the resonancethe ideasimilartothatgivenbyWang[8], butbasedontheSKastheoptimallter. Inanotherpaper, BarszczandRandall [9]showed that there are cases of gear faults, when the TSA will not give results, due to excessive frequency span between theimpulse repetition rate and the resonance frequency. In that case SK gave good results, detecting the fault several weeksbefore other compared methods. SK is currently an established method for rolling bearing fault detection, as for examplepresentedbySawalhi and Randallin[10]forboth simulated andmeasuredsignals.There are several other researchers, who refer to the SK method. One example can be the work of Bozchalooi and Liang[11],where theyproposeamorecomplexalgorithm forREBfaultdetection, basedonthewavelettransformationof thespectralresidualofavibrationsignal.TheypointtoadrawbackoftheSK,whichisthedependenceoftheestimatoronseveral parameters, like, e.g. therotational speed. Suchadependencecanbeeasilydeducedfromformula(1). Theyproposedanothermeasure, basedonthesmoothnessindex,which isless vulnerable.In this paper, a novel method is proposed for determining the optimum band for the signal demodulation. Unlike SK, themethod is not blind, as it requires certain knowledge about the sought fault. On the other hand, it is more sensitive to faultinducedcomponentsinthe vibrationsignal.Thepaperisorganizedasfollows:Section2pointsouttheproblemoftheoptimalbandselectionfortheamplitudedemodulation. The inuence of bandwidth selection, as well as of the center frequency selection is demonstrated. Section 3reviews kurtosis-based techniques applied in diagnostics of rotating machinery, with the emphasis on the Fast Kurtogram.Next, the limitations of these methods which authors have experienced in their research are listed, and a novel method fortheoptimal bandselectionispresented. Section4illustratestheperformanceof thenovel methodonbothreal andsimulateddata. Ineachcase, theresultsarecompared with theFastKurtogram-indicatedband.Throughout the paper, the authors take advantage of the narrowband amplitude demodulation technique with the use ofthe Hilbert transform and analytic signals. The scheme of the method is described by Ho and Randall [12], whereas a thoroughmathematical background is given by Jab"on ski in [13]. Note that the method preserves the spectrum resolution regardless ofthe selected bandwidth. Furthermore, all baseband spectra throughout the text were calculated with maximum resolution.2. The frequency band selection problem2.1. OverviewGenerally, natural frequencies of rolling element bearings are assumed to be most likely found somewhere in the rangefrom 5 to 20 kHz, depending mainly on the structure of the casing [14]. Although the data processing algorithms in largecommercial diagnostic systems are often constrained to demodulate vibration signals in a single multi-kHz band, includingsuchaninterval for theenvelopeanalysismakesthetechniqueineffective, becauseit introducesanumber of otherfrequency components, masking the faulty-bearing-induced signal. The frequency band selection problem is understood asthe selection of optimal center frequency and bandwidth dyad.2The practical concern in bearing diagnostics is that any of2Athoroughtheoreticalexplanationoftheproblemintermsofthespectralkurtosisisgivenin[6]. However, asitwillbeillustrated, thepaperpresentsadifferent applicationofkurtosisinspectralanalysis.T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011)431451 432thesetwoparametersishardlyeverknownapriori. Furthermore, inmanycasesthesoughtimpulsesaremaskedbyastrongnoise or othersignal componentsexcitedduring normal operation.In many band-selecting algorithms, e.g. in the Kurtogram, the emphasis is rst put on the selection of the center frequency,andconsecutivelyanoptimalbandwidthisselectedtomaximizethekurtosisbybalancingthelengthofimpulseresponseagainsttheamountofmaskingsignaltransmitted.Inthisapproach,itisactuallybesttohavethebandaswideaspossible,without introducing extraneous signals, so as to make the impulse responses as short as possible, and increase the kurtosis.Inthepaper, theauthorspresent anovel techniquefortheoptimal bandselection, whichtakesadvantageof themechanical parameters of elements being diagnosed. In the algorithm, the bandwidth is set as a rst parameter, and thenthe optimal center is chosen. The bandwidth is calculated proportionally to the sought center frequency. It is chosen shortenough to avoid extraneous signals, yet long enough to assure that damping is higher than the rate of decay of the impulseresponses. Although the bandwidth may be further optimized by other techniques (e.g. genetic algorithms), the potentialbenetisalwaysaccompaniedwithextracomputationalburden, andwasnotinvestigatedinthepaper.Ultimately, thepresentedtechniqueallowsdetectionofsuchfrequencyband, forwhichtheenvelopespectrumcontainingthesoughtcharacteristic frequency has thehighest signal-to-noise ratio.Asanexampleofthefrequencybandselectionproblem, arealvibrationsignalcontainingarollingelementbearingouter race defect-induced component is examined. For demonstration purposes, a relatively clear test rig signal recorded intheAGH University ofScience andTechnology wasused.Fig. 1presentsthetest rigand theREBouter racedefect.Therollingelement bearingcharacteristics arepresentedinTable 1.2.2. SelectionoftheoptimalbandwidthThebandwidthcanbeselectedfromaslowasfew spectrallines(inthecaseofadiscretespectrum)uptotheentirebaseband (in the latter case, the center frequency would have to be selected in the center of the frequency range). Selectingtheoptimal bandwidth isacompromise betweenanumber of aspects,three of which arediscussedbelow.The rst aspect suggesting narrowing of the bandwidth is related to the clearest demodulated signal possible, i.e. thesignal which includes only the sought components. Theoretically, if such a frequency band is selected, which includes onlyasinglecarrierfrequencymodulatedbyaREBcharacteristicfrequency, theresultantoutputwouldbeasingle, fault-inducedcharacteristicfrequencyofanunequivocal diagnosticinterpretation. However, thisisnotpossibleinpracticaldiagnosis, because:REBfaultsinduceimpulsesinthetimesignal,energyofwhichbecomesdistributedoverawiderangeoffrequenciesafter transformingintothe frequency domain,subsequently lteredby alltheresonances;Fig. 1. Testrigandthevisible REBouterracedefect(cagepurposelyslightlyrotated).Table 1Studiedrollingelementbearingcharacteristics.Parameter ValueNumberofrollingelements 14Shaftrotation speed 25.00 HzBalldiameter 7.49 mmPitchcirclediameter 45.4 mmLoadangle 01Samplingfrequency 24 kHzBPFO 144.4 HzBPFOx2 288.8 HzBPFOx3 433.2 HzT.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011) 431451 433generally, foralgorithmsselectingoptimal centerfrequencyforagivenbandwidth, selectionof smallerbandwidthdramatically increases time consumption,since itrequires extra iterations.Thesecondaspect suggesting widening of thebandwidth isreferred to thelargestamount ofdata to beprocessed:arelativelywidebandwidthassuresahighpercentageof thedefect-inducedsignal energytobeincludedintheenvelope analysis, butlargernoise willbeincludedaswell;incaseofmachinetrainswithdiverseREBs, awidebandwidthallowsalargerquantityofbearingstobediagnosedsimultaneously, which inmany industrialapplications isof utmostimportance.The third aspect is referred directly to the diagnostic requirements for REB regardless of the spectral technique used (e.g.linear spectrum, order spectrum or envelope spectrum). As stated in [12], and endorsed by the authors own experience aswell: Inbearingdiagnostics, itisoftendesirabletobeabletodetectuptothethirdharmonicof thebearingdefectfrequency in the envelope spectrum. However, this is not always possible as the higher harmonics may have decreased inamplitudetosuch astagethat theyarebelowthe backgroundnoise.Following selection method of the optimal bandwidth for the envelope analysis, proposed by authors, will take all thethree aforementioned aspects into account. In order to support the idea, Fig. 3 illustrates the inuence of the width of theselected band on the resultant envelope spectra used for REB diagnosis. The optimal center frequency (found empirically apriori) is xed, and equal to 4 kHz, while the bandwidth varies from 200 Hz to 4 kHz (bandwidths outside these limits donotbringany extrainformation). Allgures aregeneratedfrom a10 ssignalwith amaximum resolution (0.1 Hz).In Fig. 2a (BW=200 Hz), only the rst harmonic of the BPFO is visible, plus a signicant noise. In Fig. 2b (BW=400 Hz),two harmonics of BPFOs are clearly visible, with reduced noise level. In Fig. 2c (BW=500 Hz), three rst harmonics of BPFOarevisible. Thenext case, Fig. 2d(BW=1000 Hz) enables distinguishinguptorst veharmonics of theBPFO. ForBW=3000 Hz(Fig. 2e), thoughthespectrumdisplaysuptoelevenharmonicsof theBPFOabovethenoiselevel, itisconsidered by the authors to be less clear than Fig. 2c, since it requires additional zooming for accurate interpretation anddoes not contain additional information about the fault. In the last case, Fig. 2f (BW=4000 Hz), though the amplitude scaleremained the same, the spectrum brings no diagnostic information, due to masking components and a high noise energy.From theplotsabove, following statementscanbemade:Selectingabandwidthlessthanthefundamentalsoughtfrequencyispointless, sinceitcouldnotbelocatedonthefrequency axis.For a modulated signal, a certain bandwidth threshold exists, above which the resultant spectrum is useless in terms ofREB diagnosis, because: (i) characteristic frequencies become too close to each other for a human eye to identify and (ii)thelteringprocess letsintoomany superuous components.Selectingabandwidth, whichincludes from1to5harmonic lines of thecharacteristic frequencyis legibleandsignicant fordiagnosis.Thus, for the narrowband envelope analysis, the authors propose to select a bandwidth, which includes the 3rd harmonicof the characteristic frequency. For instance, if (for a given rotational speed) the BPFO equals 144.4 Hz, it seems rational toselectbandwidth43,U144.4 Hz,e.g.equal to500 Hz.Fromtheabovestatementanotherconclusionof utmostimportanceintermsof practical applicationof diagnostictechniques canbemade.For machine trains with high gear transmission rates (e.g. wind turbines), a simultaneous use (i.e. with the same centerfrequency and the same bandwidth) of the narrowband envelope analysis for diagnostics of all rolling element bearings inthe machinetrainis inefcient (or evenimpossible). Since the exact bearinginformationis usuallyprovidedbyamanufacturer, authorssuggestperformingseparatenarrowbandenvelopeanalysisforbearingswithrelativelydiversecharacteristic frequencies.Thecurrentsectiondealt withtheproblemof selectingthebestbandwidthforthenarrowbandenvelopespectralanalysis. Thenext sectiondealswiththesecondbandparameterforthenarrowbandenvelopeanalysistheoptimalcenter frequency.2.3. SelectionoftheoptimalcenterfrequencyIf no other technique is available for the selection of the center frequency, the experimental (i.e. trial-and-error) methodmaybetheonlyoption. Fig. 3illustratestheinuenceof thecenterfrequencyselectiononthenarrowbandenvelopespectrum, for the same test rig data used in the previous section. As proposed by authors, the bandwidth is held constant500 Hz (for the studied bearing, the bandwidth of 500 Hz ensures covering the rst three harmonic lines of thecharacteristic BPFO), while the center frequency varies from 2 kHz in Fig. 3a up to 5 kHz in Fig. 3f (as the frequencies below2 kHzand above5 kHzwerepreviously foundbyauthors tobeinsignicant andareomitted fortextconciseness).T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011)431451 434Clearly, therst threeharmonicsof theBPFOarebest visibleontheresultant envelopespectrumfor thecenterfrequency equal to 4 kHz (Fig. 3d). For center frequencies 2 and 5 kHz, the amplitudes of the BPFO harmonics are close tothe noise level. As the center frequency approaches 4 kHz, rst three harmonics of the BPFO protrude more above the noiselevel. Further experimental investigation of the optimal center frequency around 4 kHz (not shown in the gure) broughtonlyaminor enhancement ofthe nalnarrowband envelope spectrum.Since the narrowband envelope spectrum for parameters BW=500 Hz, and CF=4 kHz very clearly displays the rst threeharmonics ofthesought BPFO,these parameterscanbeconsidered closeto optimal forthemethod.Itisworthmentioningthat(evenforatestrigvibrationsignal)shiftingthecenterfrequencyabout1 kHzfromtheresonant frequencymaydisqualifythenarrowbandenvelopespectrumintermsof diagnosticsignicance. Moreover,shiftingevenby500 HzmaycauseasignicantdegradationofthenarrowbandenvelopespectrumintermsofBPFOsdisplay3.Inthestudiedexample, theempirical methodwasabletocorrectlyselectoptimal parametersforthenarrowbandenvelope analysis. However, it is by no means a practical and efcient tool under the majority of industrial circumstances.Firstly, to a degree it is based on luck. Secondly, it requires a code modication, which makes it often unfeasible to use on-site(orforauserwithoutadvancedprogrammingknowledge).Finally,foracomplexdiagnosticandmonitoringsystemwith a large number of vibration channels, it is burdened with unaffordable time consumption. Thus, practical utilizationofamplitudedemodulationtechniquesforREBdiagnostics, includingthepresentednarrowbandenvelopeanalysiswiththeuse ofthe Hilberttransform, callsforanautomatic selection ofthe optimal band.0 50 100 150 2000123x 10-3BW = 200 HzAmplitude [g]0 100 200 300 4000246x 10-3BW = 400 Hz0 200 400 6000246x 10-3BW = 500 HzAmplitude [g]0 500 1000024x 10-3BW = 1000 Hz0 1000 2000 300000.511.5x 10-3BW = 3000 HzAmplitude [g]Frequency [Hz]0 1000 2000 3000 400000.511.5x 10-3BW = 4000 HzFrequency [Hz]BPFOBPFOx2BPFOBPFOx2BPFOx3BPFOBPFOx2BPFOx3BPFOx4BPFOx5BPFO ...BPFOx11BPFOFig. 2. From (a)(f): inuence of the BW selection for the optimal center frequency. N.B.: In practice, it is normal to display the range containing the rst35harmonics,regardlessofthe bandwidthtakenforcalculations.Theresolutionofall spectraisequalto 0.1 Hz.3Forajetengine vibrationsignal(presented inSection4.3), shiftingCF byaslittle as100 Hzpreventeddetection ofthe BPFOsignalcomponent.T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011) 431451 435The next chapter starts with a review of the techniques employing kurtosis-based estimators, of which the Fast Kurtogramproposed by Antoni in [6] deserves special attention. Nonetheless, the authors own research has shown that these methods arecharacterized by certain limitations in the optimal band selection. A study of the kurtosis estimator in the process of amplitudedemodulation had led authors to certain observations resulting in the proposal of a novel method for the optimal band selection.3. Proposal of a novel method for band selection and comparison with the Kurtogram3.1. SpectralkurtosisFDKandSKAmong other statistical moments, kurtosishas proven itselfto be useful indetection of nonstationary components insignals. This observation was applied in the eld of rotating machinery diagnostics, in order to detect transient impulsivecomponentsgenerated, forinstance, byfaultyrollingelementbearingsorgears. Suchtransientsareexcitedrapidlybyimpact forces and terminated by a machine assemblys damping. These forces excite resonant responses, the amplitudes ofwhich aremodulatedbythe impactsrepetition rate, creating componentsperiodically presentinthe signal.When a digital Fourier transform (DFT) of a signal is calculated, the low harmonics of this frequency are too small to bemeasured, andthehigherharmonicsaresmearedbecauseof randomvariationintheperiodicity. However, asDwyerillustratesin[3], itcanberecoveredbycalculatingthekurtosisof itsspectral amplitudes4fromanumberof signalsconsecutiverealizations, accordingto formula(2):FDKf 1=M_ _ Mi 1 Xf 41=M_ _ Mi 1 Xf 2_ _22whereMisthe number oftime segments,X(f)thespectralvalue ofthereal part atfrequencyf.0 200 400 60000.020.04CF = 2000Amplitude [g]0 200 400 600012x 10-3CF = 30000 200 400 6000123x 10-3CF = 3500Amplitude [g]0 200 400 6000246x 10-3CF = 40000 200 400 6000123x 10-3CF = 4500Amplitude [g]Frequency [Hz]0 200 400 600024x 10-4CF = 5000Frequency [Hz]BPFOBPFOx2BPFOBPFOx2BPFOx2BPFOBPFOx3BPFOBPFOx2BPFOx3BPFOBPFOx2BPFOx3BPFOx1Fig. 3. Inuenceofthe centerfrequencyselectionforaconstantbandwidth(500 Hz),from (a)CF=2000 Hzto(f) CF=5000 Hz.4In [16], Otonello and Pangan proposed to compute the FDK of magnitudes of the DFT foreach frequencyrather than of real and imaginary parts.T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011)431451 436The idea of the FDK calculation is illustrated in Fig. 4a. Proposing the frequency domain kurtosis (FDK) estimator, Dwyerclaims that for signal components with non-Gaussian amplitude distributions (i.e. components, amplitudes of which varyovertime), thekurtosisvaluewillbedistinctlybiggerthanforcomponentswithGaussianprobabilitydistribution(i.e.components, whose amplitude variations are perceived as an innate feature of real processes). Nevertheless, as presentedinFig. 4a, thisFDKestimatorrequiresaDFTcalculationsforanarbitrarynumberoftimedatasegments. Consequently,signicantvaluesoftheFDKestimatorproposedbyDwyerrelyontheselectionofapropertimesegments lengthandposition,which unfortunately isunknownapriori.A genuine milestone in the development of kurtosis-based estimators was taken by Antoni in [5], where he proposed aformalizeddenitionofspectralkurtosis(SK)anditsSTFT-basedestimator.ThegoalofSKistheautomaticdetectionoffrequencybandscontainingnonstationarysignalcomponents. DifferingfromDwyersapproach, inwhichhecalculatedkurtosis of a number of realizations of particular frequencys amplitude, Antoni calculates kurtosis of the complex envelopeoflteredtimesignals. TheideaisillustratedinFig. 4b. Practical considerationsofhismethod, hadledAntoni totheconcept of the Fast Kurtogram, as a tool using a lter bank approach [6]. Compared with the original Kurtogram, it requireslessCPUtime, butgives approximateresults.The Fast Kurtogram is a colormap of kurtosis values calculated for an array of frequency bands covering the entire basebandin a predened manner. It is conveniently presented on a plane, where the horizontal axis represents frequency, the verticalaxis represents the number of intervals into which the frequency baseband is divided, and the third dimension the color scaleFig. 4. Comparisonof algorithms: (a) FDK, (b) SK-Fast Kurtogramand(c) theproposedmethodof theProtrugram. (Notethat intheKurtogramsalgorithm, thoughthecomplexenvelope presumablymeans amplitudeandphase, thephaseisnot usedfor thekurtosis calculations, onlytheamplitude. In fact, it is convenient to use the power spectrum values (amplitude squared or sum of squares of real and imaginary values) directly, as thisrepresents the second moment, and only has to be squared to give the fourth moment. The amplitude would normally be calculated by taking the squarerootofthe squaredamplitude.)T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011) 431451 437 represents the kurtosis value of the envelope signal for each frequency bandwidth at each center frequency. Eventually, theidentication of the optimal band is treated as a prerequisite for a narrowband amplitude demodulation [1].Though the Fast Kurtogram is generally procient in localizing hidden nonstationarities, the authors have experiencedthat it does not cope well with signals of composite, frequently of randomly-impulsive nature, where the sought transientsignal isrelativelylow. Suchsignalsarecharacteristic, forinstance, forfaultyREBincomplexmachinetrains(e.g. jetengines) andhighlyenvironmentallyaffectedsignals (e.g. printingaggregates, etc.) Inthesecases, theKurtogramisburdened with certain drawbacks. The issue of the Kurtograms vulnerability to random extraneous signals is illustrated indetails in another authors publication [15]; however, the main point is recovered in the paper to give a complete picture.Theselectedexamples aim topresent Kurtograms limitations, which maybeovercomeby thepresentednovel method.Firstly, it tends to show a number of ambiguous pseudo-optimal frequency bands for demodulation, which inevitably resultin confusing the user. Moreover, it is prone to point a band with high kurtosis as optimal, but possibly being incorrect. This is sobecause the Kurtogram calculates each kurtosis value from a time signal generated from a band-pass ltered signal, and showsrelatively high values for time signals containing impulses of any kind, without source identication.Secondly, according to Antoni, the SK of the process x(n), with an additive noise b(n) is highly sensitive to the noise level [6]:Kxbf Kxf 1rf _ _23where r(f) is the noise-to-signal ratio as the function of frequency.On one hand, SK is able to detect a fault, even when the overall r ratio is high. It is only required that a certain band ispresent, in which r(f) is low enough. On the other hand, formula (3) states that SK decreases rapidly with the noise growth.As an attempt to overcome aforementioned concerns, the authors are proposing a novel method for the optimal bandselectionfor theamplitudedemodulation, takingadvantageof kurtosis values of thenarrowbandenvelopespectralamplitudes. ThelatterapproacheventuallyenablesidenticationofmodulatingsignalswithmuchlowerSNR, andalsoprovides a potential for differentiation between high kurtosis values caused by modulating signals and high kurtosis valuescausedbyother spectralcomponents.3.2. ProposalofanovelmethodfortheoptimalbandselectionAs mentioned before, the Kurtogram is based on kurtosis values calculated from envelopes of modied time signals. Onthe other hand, the proposed method takes advantage of kurtosis values calculated in the frequency domaindirectly fromtheamplitudesofthespectrumofthenarrowbandenvelopesofasignal, asillustratedinFig. 4c. Followingdiscussionrevealsthebenets ofthelatter solution.In statistics, the kurtosis is dened in words as the peakedness or atness of the graph of a frequency distribution [17].For a sample of N values, it is calculated as a biased estimator from the sample fourth and second moments, whereas for apopulation, it iscomputed as an unbiased estimator using cumulants. Although the engagement of the signal processingtheory puts additional constrains on the kurtosis denition, the authors deliberately use the most basic kurtosisdescription(i.e. fourthstandardizedmoments) inorder tokeepthe clarityof the presentedalgorithm, andnot tooverwhelm thepaper withextra equations.Since thealgorithm compares kurtosis-basedoutputsrelatively, theauthor suggeststouse thefollowing denition:K mksk , 4where mk is the fourth moment about the mean value of the number sequence, while skis the standard deviation raised tothefourthpower, whichmeasurestheprotrusion(orthepeakedness)ofasignal. ForavectorX={X1,...,XN}, thekurtosisK(X)canbecalculatedas5KX m4s4 1=N_ _ Ni 1 XiX41=N_ _ Ni 1 XiX2__ _4 1=N_ _ Ni 1 XiX41=N_ _ Ni 1 XiX2_ _2 1=N_ _ Ni 1Xi 1=N_ _ Ni 1 Xi_ _ _ _41=N_ _ Ni 1Xi 1=N_ _ Ni 1 Xi_ _ _ _2_ _25Considering Fig.5,kurtosisasameasure of theprotrusion ofa signal:showsthe highestvalue ifallbutonenumbers inthesequencearethesame (Fig. 5b)for a given number of components sticking out from the sequence, it reaches a maximum value when the dispersionofthe amplitudesislargest (Fig. 5e andf).Note that these number patterns may actually correspond to possible shapes of narrowband envelope spectraillustratingthepresenceof amodulatingsignal. Forinstance, inthepresenceof aREBouterracefault, itisanatural5Alternatively, a constant value of 3 may be subtracted from the resultant value; however, as long as consistency in calculations is preserved, in caseofcomparativeestimatorsofvibrationsignals,thissubtractionis irrelevant.T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011)431451 438behaviorofharmonic linesofcharacteristic frequencies (e.g.1x,2xand3xBPFO)tohavedescendingamplitudes,whichcorrespondstoFig. 6f. Foraspectrumoftheenvelopecontainingthreeclearconsecutiveharmonicsofacharacteristicfrequency, thisisthecasewhenkurtosis showshighest value.Thoughtheauthorsdohaveexperiencewithinnerracefaultsaswellasrollingelementfaults, theseexampleshavebeen omitted due to the text conciseness. Obviously, the appearance of sidebands decreases the kurtosis comparing to pureharmonics, but the presented estimator puts more strength to the overall signal-to-noise ratio, and it will still point out anoptimal band, where harmonics plus sidebands are clearest visible, whichdoes not have tocorrespondtohighestamplitudevalues.Theauthorsbelieveasingle-lineexamples(e.g.outerraceharmonics)betterdemonstratethemainideaof theprotrugram. Thus, kurtosiscalculatedfromamplitudesof anumberof narrowbandenvelopespectramayindicatewhichspectrumcontainsdetectablespectral components. Consequently, thecorrespondingcenter frequencyindicates theresonantfrequency.Indeed, theabovementionedobservationisthekeyideaoftheproposedmethod, whichdisplayskurtosisvalueofanumber of calculated narrowband envelope spectra as a function of the center frequency (see Fig. 4c). For each subsequentenvelope spectrum, the center frequency CF is shifted by a predened step (away from 0 frequency), while the bandwidthBW is held constant. According to statement made in Section 2.2, the bandwidth is chosen slightly more than three times0 100 200 300 400 5000510Kurtosis=NaNAmplitude0 100 200 300 400 5000510Kurtosis=4980 100 200 300 400 5000510Kurtosis=248Amplitude0 100 200 300 400 5000510Kurtosis=2770 100 200 300 400 5000510Kurtosis=164Sample indexAmplitude0 100 200 300 400 5000510Kurtosis=230Sample indexFig. 5. Kurtosisvaluesforpatternsofnumbersgeneralizing afewpossible shapesof envelopespectra.0 5 10-2-1012Time [s]Amplitude [g]5.1 5.15 5.2-2-1012Time [s]Fig.6. (a) Time viewof thetestrigsignaland(b) timeview ofthe testrigsignalzoom.T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011) 431451 439the sought characteristic frequency. In this way, a pair of necessary input parameters to the narrowband envelope analysismaybeobtained.The narrowband envelope analysis requires two input parameters, the bandwidth and the center frequency. In contrarytotheKurtogram, whichconsiders avarietyof different bandwidths andcenter frequencies, thenovel methodwasdeveloped from an idea of setting the bandwidth to a xed value, and search for the optimal center frequency. Since thegoal of the method is eventually the same as of the Kurtogram, and the method is based on kurtosis values as a measure oftheprotrusion ofthe signal, authorsproposetoname ittheProtrugram.Duetothealgorithmconstruction, for relativelystrongmodulatingsignals, thenewmethodhasthepotential ofpointingtheoptimal center frequencyviaanabsolutemaximumontheProtrugram, withtheaccuracyequal totheresolutionoftheoriginal spectrum. However, forreal, compoundvibrationsignalswithrelativelylowSNR, thecenterfrequency may be indicated by a local maximum. Nevertheless, authors presume that the energy of the modulating signalisalwaysdistributedaroundthecarrierfrequency. Thus, astheCFisshifted, itgraduallyapproachestheoptimalvalue(enhancingthevirtualprotrusionofcharacteristicfrequenciesabovethenoiselevel), andretreatingfromitafterwards(protrusion deteriorates). Consequently, center frequencies of modulating signals are displayed on the Protrugram by hill-like shapes. On the other hand, the remaining envelope spectral components are assumed to be introduced by individualspectral component. Thus, as the CF is shifted, their maxima appear and disappear suddenly. Consequently, they may bedifferentiatedfrommodulatingsignals-indicatedoptimal center frequenciesontheProtrugrambyrapidsteepedges,which callsfor additionalpostprocessing. Thesestatements willbediscussed furtheronthe basisofexemplary signals.4. Case studiesAuthors present ve examples of the application of the Protrugram in detection of the optimal frequency band for theamplitude demodulation. In each case, the results are compared with a Fast Kurotgram-recommended band. As it will beshown, forsignalsincludingrelativelystrongfault-inducedsignal components, bothmethodsworkcorrectlyandgivesimilar results. However, in the case of relatively weak modulating signal, i.e. relatively small signal-to-noise ratio, the FastKurtogram is unable to indicate the optimal band for demodulation, whereas the Protrugram enables a successful selectionoftheoptimal band.Casestudies include:testrig vibrationsignal (introducedinSection 2.1);synthetic computer-generatedsignal simulating aREBdefectwith relatively highSNR;synthetic computer-generatedsignal simulating aREBdefectwith relatively lowSNR;synthetic computer-generatedsignal simulatingaREBdefect withrelativelyhighSNRandanimpulseresponse,simulating arandompeak;vibrationsignal froma jetengine withabearing outerracedefect.4.1. TestrigRecalling, the studied test rig signal is a 10-s sample recorded with a sampling frequency 24 kHz, containing a REB outerrace defect-induced component. Fig. 6a and b present the time signal. Fig. 7a and b illustrate the one-sided linear spectrumof the signal, and the dB spectrum. Figs. 810 illustrate the Fast Kurtogram, the Protrugram and the narrowband envelopespectra,respectively.The time view of the signal (Fig. 6) does not unravel any REB fault symptoms. The linear spectrum (Fig. 7a) shows frequencycomponents lessthan 2 kHz. The dB spectrum (Fig. 7b) shows a gradual decrease of spectral amplitudesfor frequencies lessthan 3 kHz, then a steady 85 dB level to 5 kHz. After that, a 95 dB amplitude level is observed up to the Nyquist frequency.Fig. 10 shows narrowband envelope spectra calculated for optimal parameters indicatedby bothmethods, theProtrugramandtheFastKurtogram.Inthelattercase, allharmonicsoftheBPFOprotrudehigherabovethenoiselevel;however, theProtrugramwas able toindicatea band, for whichthefundamental BPFOhas got higher amplitude.Nevertheless, bothmethods showverylegiblenarrowbandenvelopespectradisplayingconsecutiveBPFOs, enablingsuccessful REBdiagnosis.AsindicatedinFig.4c,thecalculationof theProtrugram requiresadditionalparameterthesizeofthestep ofscanning,loosely, how much the central frequency is shifted on the frequency axis after each iteration. Step sizes of 1000, 100 and 1 Hzwere investigated on a test rig signal. The nest accuracy of the carrier signal frequency was found to be about 4068 Hz. Fig. 7aand b clearly show that the frequency content of the signal is dominated by frequencies about 1 and 2 kHz, which are believedto cause sharp peaks on the Protrugram. Fig. 11 presents Protrugrams calculated for three step sizes listed above.TheTable2presents computationtimes for eachstep. Computations presentedinTable2wereperformedona2.40 GHz IntelsCeleron with 632 MB RAM, in Matlabsversion 7.0.1.24704 (R14) Service Pack 1 on a 10 s time signal withsampling frequency 24 kHz. The result from the rst row, for the step size equal to 1000 Hz is the fastest; however, after acloselookonthegraphsitcanbearguedthattheresultiscoincidentlycorrectbecauseiftheiterationwasstartedatadifferent frequency, then the kurtosis would not indicate optimal center frequency. It is therefore necessary to increase theT.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011)431451 440numberofiterations, asshowninthesecondrowofthetable. Inthenalstep, i.e. withthemaximumresolution, thenumberofiterationswasfollowedbyarapidincrementofthecomputationaltime.Moreover,theimprovementsoftheresultantCF seeminadequate tothe extratime consumption.Taking into account the time consumption for each step and corresponding results of the optimal center frequency, itseems reasonable to use a step size of the order of about 100 Hz, if applicable. Nevertheless, more powerful computationalmachines allowbetteraccuracy.0 2000 4000 6000 8000 10000 1200000.10.2Amplitude [g]0 2000 4000 6000 8000 10000 12000-150-100-500Amplitude [dB]Frequency [Hz]Fig.7. (a) Linearfrequencyspectrumof thetestrigsignal.(b) dB-scalefrequencyspectrumofthe testrigsignal.frequency [Hz]level kfb-kurt.2 - Kmax=1.2 @ level 4, Bw= 750Hz, fc=4125Hz 0 2000 4000 6000 8000 10000 12000011.622.633.644.655.666.677.6800.20.40.60.811.2Fig. 8. The Fast Kurtogram of the test rig signal. (All Fast Kurtogram plots were generated with the use of the code available in [18], with the parameterslterbankandclassical.)0 2000 4000 6000 8000 10000 1200005001000KurtosisFrequency [Hz]Fig.9. Protrugramofthe testrigsignal.BW=500 Hz,indicatedCF=4068 Hz.T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011) 431451 4414.2. SimulatedsignalAnothersignal, whichdemonstrateslimitationsoftheFastKurtogram, isasimulatedsignal, whichwaspreparedinMatlabs. The signal was sampled during 10 s with a frequency of 25 kHz (data packets 250 thousand samples each). Thesignalconsists offollowing components:threesinusoids(withrandomphases), representingatwo-shaftmachine(35and11 Hz)withameshingfrequency(455 Hz),comprising dominating signalcomponents(amplitudes 1.0,0.8and0.3,respectively);0 100 200 300 400 500024x 10-3Frequency [Hz]Amplitude [g]0 100 200 300 400 500024x 10-3Frequency [Hz]BPFOBPFOx3BPFOx2BPFOBPFOx2Fig.10. Narrowbandenvelopespectrumforindicatedoptimalparameters:(a) Protrugramand(b) FastKurtogram.0 2000 4000 6000 8000 10000 120000200400Kurtosis0 2000 4000 6000 8000 10000 1200005001000Kurtosis0 2000 4000 6000 8000 10000 1200005001000KurtosisFrequency [Hz]CF = 4500 HzCF = 4100 HzCF = 4068 HzFig. 11. Protrugrams forconstantBW=500 Hzandvaryingstep size:(a) 1 kHz, (b)100 Hzand(c) 1 Hz.Table2Computationaltimes foreachstep sizeofthe Protrugram.Frequency step size (Hz) Indicatedbest f_center (Hz) Maximum kurtosis value No. of iterations Computing time (s)1000 4500 682.6 12 0.6100 4100 761.3 118 3.41 4068 767.6 11750 322.5T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011)431451 442modulating signal component, simulating forinstance a bearingrace fault.The signalwas composed of impacts fromexponentiallydecaying4 kHzsinecarrier frequencywithinitial amplitudeof 0.1, andtimeconstant of 2 ms; theimpacts repetition ratewas124 Hz with a1% randomjitter;stationaryrandomGaussiannoise; energyof thenoisewaschangedinordertodeterminethedependencyof thedetection method.The simulated signal is examined in three variants, high signal-to-noise ratio, low signal-to-noise ratio, and high signal-to-noiseratio with arandom impulse.Therst caseillustrates thecapability of theFast Kurtogram topoint theoptimalbandaccurately, whereasthe lattertwo casesillustrate itsvulnerability toSNR andrandom impulses,respectively.4.2.1. Highsignal-to-noiseratioIn the rst test, the noise component had the variance of 0.01. The overall signal with its components is presented inFig. 12.Fig. 13 presents the amplitude spectrum of the entire signal, together with PSDs of the signal and its components. It isclearlyvisiblethat atthe resonance frequency,theSNR isapproximately threeorders ofmagnitude.The simulatedsignal was processedwithboth, the Fast Kurtogramandthe Protrugram. Results of analysis arepresented in Figs. 14 and 15, respectively. The Fast Kurtogram detected the maximum of SK at 2344 Hz with the bandwidthof 1562 Hz (and a relatively high SK value equal 4.6), which do not correspond exactly with the original carrier frequency.TheProtrugramdetectedaveryclear maximumat 4 kHz, whichis inlinewiththecarrier frequency. Theselectedbandwidth was400 Hz (slightly morethan3x possibleBPFO).Bandparametersreturnedbybothinvestigatedmethodswasusedtoobtainnarrowbandenvelopespectra. Thesespectra are compared in Fig. 16. Both spectra show presence of the fault repetition rate, together with its harmonics. Thus,both methods wereabletodetectthefault signature.NotethattheFastKurtogramwasabletoindicateaband, whichenabledidenticationof anumberof harmonics(actually, up to the 11th harmonic) of a possible BPFO, whereas the Protrugram indicated band, which identied only rsttwo harmonics. However, for the implemented algorithmof amplitude demodulation, amplitudes of characteristicfrequencies inthe caseof the Protrugram arealmost four thousand times higher thanin thecase of theFast Kurtogram.4.2.2. Lowsignal-to-noiseratioInthesecondtest, thenoisecomponenthadmuchhighervarianceof0.25.TheoverallsignalwithitscomponentsispresentedinFig. 17.Fig.18presentstheamplitudespectrumofthetotalsignal, togetherwithPSDsofthesignalanditscomponents.Noresonanceisobservableonthespectrum. ComparisonofPSDsshowthatthepowerofthemodulatingsignal(i.e. fault-induced)issmaller thanthenoise,evenatthe resonance frequency.Thesimulatedsignal wasprocessedagainwithbothmethods, theFast KurtogramandtheProtrugram. ResultsofanalysisarepresentedinFigs.19and 20,respectively.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-202Amplitude [g]0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-101Amplitude [g]Time [s]Fig. 12. (a) A part of the 10 s simulated signal in the case of high SNR and (b) its components. The noise is relatively small and impacts are visible. Notedifferent timerangeon bothplots.T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011) 431451 443This time, the Fast Kurtogram was not able to detect a valid band. The Protrugram, on the other hand, was still able todetectamaximumat4 kHz, thoughtheamplitudeof themaximumwasmuchlowerthaninthehighSNRcase. TheparametersfortheProtrugram werethesame asinthepreviouscase(BW=400 Hz, step=1 Hz).Bandparametersreturnedbybothinvestigatedmethodswereusedtoobtainenvelopespectra. Thesespectraarecompared inFig.21.Thistime, onlytheparametersreturned bythe Protrugramyieldedacorrect envelope spectrum, inwhichthefaultfrequencycanbeclearlydetected. TheenvelopespectrumobtainedthroughtheKurtogramshowafewspectrallines,butunrelated tothesought fault.The ability to detect the fault was further investigated in consecutive simulations. Table 3 presents results of SNRs andlimit noise variances for both methods. Maximum noise energy presents maximum variance of the noise signal, at whichthe method was ableto detect the fault. SNR values were calculated from the time signals, so they include all frequencycomponents.SNR asa function offrequency canbeestimated fromFigs.13and 18.4.2.3. Highsignal-to-noiseratioplusarandomimpulseInthelasttest, thenoiselevel iskeptlow, but anextraimpulseresponsehasbeenaddedtothesignal, whichisobservableinFig. 22. Thespectraareomitted, sincevirtuallytheylookthesameasintherstsimulatedsignal test.Comparing with Figs. 14 and 23 indicates that the presence of a relatively low energy impulse response in a signal may ruinthe Fast Kurtogram analysis. On the other hand, Fig. 24 illustrates that the Protrugram output has not changed signicantlyfrequency [Hz]level kfb-kurt.2 - Kmax=4.7 @ level 3, Bw= 1562.5Hz, fc=2343.75Hz0 2000 4000 6000 8000 10000 12000011.622.633.644.655.666.677.681234Fig. 14. Result of the Fast Kurtogram analysis of the simulated signal. The maximum around 2.3 kHz at the level 3 is clearly seen, which is close to theoriginalcarrierfrequencyof4 kHz.0 2000 4000 6000 8000 10000 1200010-410-310-210-1100Frequency [Hz]Amplitude [dB]0 2000 4000 6000 8000 10000 1200010-610-410-2100102Frequency [Hz]Amplitude [dB]Total SignalNoiseModulating SignalFig.13. (a) Amplitudespectrum(dB) ofthesimulated signaland(b)PSDsof thesignaland itscomponents.T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011)431451 444compared to the impulse-free signal. Fig. 25 shows envelope spectra for indicated optimal bands, which clearly show thatthis time the FastKurtogram-based analysisfails.Thus, theauthorsclaim that theProtrugram constitutes analternativeapproachto theFastKurtogram analysis, since itshows lessvulnerability to randomimpulsivenoise.4.3. JetengineThe investigation of a signal recorded from a jet engine was one of the reasons which have motivated the authors to ndanalternativemethodfor theoptimal bandselectiontotheKurtogram. Dismantlingof themachinerevealedseverebearingracefault. TheFastKurtogramdidnotyieldanysignicantband, whichcouldbeusedfordemodulationofthesignal. One reason for this was a relatively very low signal-to-noise ratio (or very high r in terms of formula (3)). Secondly,thesignal containedanumberof peaks, whichseemedtoberandomandpossiblyexcitingresonances. Tertiary, the100 200 300 400 50000.20.4Frequency [Hz]Sq.Amplitude [g]100 200 300 400 50000.51x 10-3Frequency [Hz]1x2x1x2x3x4xFig. 16. Narrowband envelope spectrum for indicated optimal parameters: (a) Protrugram and (b) Fast Kurtogram. Note necessary different amplitudeordersfora cleardisplay.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-202Amplitude [g]0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-101Amplitude [g]Time [s]Fig. 17. (a) A part of the simulated signalin the case of lowSNR and(b) its components.The noiseis relatively high and impactsarenot visible. Notedifferent timerangeon twoplots.0 2000 4000 6000 8000 10000 12000 1400005001000Frequency [Hz]KurtosisFig.15. Resultofthe Protrugramanalysis;BW=400 Hz,step=1 Hz.Themethodcorrectly indicatedoptimalcenterfrequency.T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011) 431451 445nominal speed of the engine was 8000 rpm, requiring a demodulation bandwidth over 1.4 kHz (due to a very high BPFO of1325 Hz;suchashortrepetitionratemaycausethatfault-inducedimpactdoesnotdecaycompletelybeforethenextimpact occurs), which in turn demanded a relatively accurate detection of the center frequency. Furthermore, the vibrationsignalhadalargenumberofcomponents,comingfromrotatingparts(shaftsandseveralgeartransmissions), includingblade pass frequencies of the compressor and the turbine or broadband noise from gas ow and combustion processes. Inthe presented case, various kinematic parameters of the machine were known (which is often the case encountered in thediagnostic practice). This is especially true for rolling element bearings diagnostics, since characteristic frequencies can be0 2000 4000 6000 8000 10000 1200010-310-210-1100Frequency [Hz]Amplitude [dB]0 2000 4000 6000 8000 10000 1200010-610-410-2100102Frequency [Hz]Amplitude [dB]Total SignalNoiseModulating SignalFig. 18. (a) Amplitudespectrum(dB) ofthe simulatedsignaland(b) PSDsofthe signalanditscomponents forthelowSNRcase.frequency [Hz]level kfb-kurt.2 - Kmax=0.3 @ level 1, Bw= 6250Hz, fc=3125Hz0 2000 4000 6000 8000 10000 12000011.622.633.644.655.666.677.6800.050.10.150.20.25Fig. 19. Resultofthe FastKurtogramanalysisof thesimulatedsignal withlowSNR.Themethodwasnot ableto detectthe faultsource.0 2000 4000 6000 8000 10000 1200001020Frequency [Hz]KurtosisFig.20. Resultof theProtrugramanalysisforthelow SNR.T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011)431451 446relativelyeasilyobtainedfromthe manufacturer. Another important issueconcernedsignal transmissionpath. Thevibration signal was recorded with an acceleration sensor mounted on the chassis of the engine. Thus, it was relatively farfromthe faulty bearing, so the fault signature component was weaker and the signal contained more structuralfrequencies.Fig. 26 presents the waveform of the signal and its amplitude spectra (in both linear and logarithmic scale). The signalwas sampled at 25 kHz for 10 s. The anti-aliasing lter had the cut-off frequency at 10 kHz. Even after zooming (Fig. 26a),0 100 200 3000.20.40.6X: 123.9Y: 0.4779Frequency [Hz]Sq.Amplitude [g]0 200 400 600012X: 420Y: 0.9018Frequency [Hz]Sq.Amplitude [g]Fig. 21. Narrowbandenvelopespectrumfor indicatedoptimal parameters: (a) Protrugramand(b) Fast Kurtogram. Notedifferent amplitudeandfrequencyrangesapplied fora clearerdemonstration.Table 3Parametersofsimulatedsignals.Parameter ValueHigh SNRcaseRMSof thefault signal 0.0249RMSof thenoise 0.0100SNR 6.20Low SNRcaseRMSof thefault signal 0.0249RMSof thenoise 0.2505SNR 0.0098MaximumnoiseRMS forthe FastKurtogram 0.083MaximumnoiseRMS forthe Protrugram 0.2900 2 4 6 8 10-202Time [s]6.71 6.72 6.73 6.74 6.75-101Amplitude [g]Time [s]Fig. 22. (a) A simulated signal with high SNR containing a response of a random impulse. The impulse response signature visible (almost negligible) onthe time view is marked with a dotted line. (b) Zoom on its components at the time of the impulse occurrence. Note the relatively marginal inuence ofthe impulseresponse tothe overallsignalcharacteristics.T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011) 431451 447thetimesignal showsquiteatypical vibrationsignal, withoutanysignof largeimpactsormeasurementerrors. Thespectrum (Fig. 26c) contains several discrete lines. All spectral lines were identied and found not related to bearing faultfrequency. Dominant lines around6 kHz come frombladepass frequencies inthecompressor. Investigationof thelogarithmic spectrum(Fig.26d)doesnotshow any clear,dominant resonance frequency.The Kurtogram analysis was performed and did not yield a signicant band for signal demodulation (see Fig. 27). Then,theProtrugramwasappliedandreturnedtheplotpresentedinFig. 28. Theplotcontainsamaximumwithsteepedgesbetween5.7and7.2 kHz.AsdiscussedinSection3.2,suchsteepedgesareaconsequenceofharmoniccomponentsandshould be ignored. Indeed, they were found to be caused by blade pass induced harmonic components, which can be seenin Fig. 26c. Apart from that maximum, two much smaller, yet visible local maxima around 8200 and 11 900 Hz are present.Both structures have gentle slopes and may be caused by a repetitive fault signature. These hypotheses were veried withfrequency [Hz]level kfb-kurt.2 - Kmax=2087.3 @ level 4, Bw= 781.25Hz, fc=8984.375Hz0 2000 4000 6000 8000 10000 12000011.622.633.644.655.666.677.680500100015002000Fig. 23. ResultoftheFastKurtogramanalysis. Themaximumaround9 kHzatthelevel4isindicated, whichdoesnotcorrespondtothetruecarrierfrequency(4 kHz).0 2000 4000 6000 8000 10000 12000 14000050010001500Frequency [Hz]KurtosisFig.24. Resultof theProtrugramanalysis; BW=400 Hz,step=1 Hz.Theindicatedcenter frequency=3985 Hz.0 100 200 300 400 50000.10.20.3Amplitude [g]Frequency [Hz]0 100 200 300 400 5000123x 10-3Frequency [Hz]BPFOBPFOx2Fig. 25. Narrowbandenvelopespectrumforindicatedoptimal parameters:(a)Protrugramand (b)FastKurtogram.Notedifferent amplituderanges.T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011)431451 448narrowbandamplitudedemodulationof thevibrationsignal aroundbothcenterfrequencies. Onlytherstmaximumturnedouttoberelatedtothesoughtbearingfault. Forthelatterone, asingle1000 Hzcomponent, causedbyanothersource, was observable above the noise level. The result for the 8200 Hz maximum is presented in Fig. 29a, where a BPFOcomponent isclearlyvisible.0 5 10-0.200.2Amplitude [g]Time [s]0 0.005 0.01 0.015 0.02-0.200.2Time [s]0 2000 4000 6000 8000 10000 12000 140000246x 10-3Amplitude [g]0 2000 4000 6000 8000 10000 1200010-610-410-2Frequency [Hz]Amplitude [dB]Fig. 26. Vibration signal from the jet engine. (a) Time signal. It does not show any sign of impacts. (b) Linear spectrum. Spectral lines were identied andfound notrelated tobearingfault frequency.(c)dBspectrum.It didnot showanycleardominant resonancefrequency.frequency [Hz]level kfb-kurt.2 - Kmax=0.3 @ level 1.5, Bw= 4166.6667Hz, fc=2083.3333Hz0 2000 4000 6000 8000 10000 12000011.622.633.644.655.666.677.6800.050.10.150.20.250.3Fig. 27. TheKurtogramofthe vibrationsignal fromthe jetenginewitha faultybearing. Notethatkurtosislevelsarerelativelylow.T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011) 431451 449Theenvelopespectrumcontainsthelineat thebearingracefault frequency1325 Hz. Thespectrumalsocontainsspectral components of 290.3 and 580.6 Hz (its second harmonic) as well as 1061 Hz, which were found related to meshingfrequencies in one of auxiliary gears. For comparison, the envelope spectrum for the band yielded by the Fast Kurtogram isalsopresented(Fig.29b).Itdoesnotshow anysign ofabearing fault.This example illustrates that the demodulationwiththe Protrugram-indicatedparameters shows a potential toovercome limitations encountered in demodulation with Fast Kurtogram-indicated parameters. Conducting morenarrowbandenvelopeanalysisonthesignalwiththebandwidthequalto1500 Hz,itturnedoutthatshiftingthecenterfrequency byaslowas100 Hz, totallydeteriorates theenvelope spectrum intermsof theBPFOidentication.This observationis further justiedby the fact that inthe Fast Kurtogramalgorithm, consecutive segments ofdemodulation bands for a given bandwidth are non-overlapping; thus, as the sought bandwidth increases, the accuracy ofthecalculatedcenterfrequencydiminishes.Inthecaseofthecharacteristicfrequency1325 Hz, thesmallestBWontheFast Kurtogram capable of its detection is equal to 1563 Hz (level 3), with accuracy of 1563/2=782 Hz, which was found tobeinsufcient.5. ConclusionsThepaperoriginatedfromaresearchonspectral kurtosisandtheKurtogrammethodsintermsofthenarrowbandenvelope analysis. The idea of an alternative method for the optimal band parameters selection emerged when the authorswere investigating a signal from a jet engine. The implementation of the Fast Kurtogram did not yield any signicant band,thoughdismantling ofthemachine revealedseverebearingracedefect.Authors believe that the Fast Kurtogram failed due to a number of reasons. First, the signal-to-noise ratio was relativelylow. Secondly, thesignal containedavarietyof peaks, possiblyexcitingresonances. Finally, thecharacteristicsoughtfrequencywaslarge(1325 Hz),which requiredabetteraccuracy ofthepossiblecenterfrequencies beingaccessedthanofferedby theFast Kurtogram forsuchalarge bandwidth.Analysis of practical cases, as well as experiments on the simulated signals had led to the concept of another REB faultdetectionmethod. Suchamethod, namedbytheauthorstheProtrugram, wasproposedanddiscussedinthispaper. Incontrary to the Kurtogram, the new method requires a priori knowledge about kinematics of the monitored machine, andadditional visual postprocessingtoreject discretetones. Ontheother hand, it shows asuperior detectionabilityofmodulating signals in presence of higher noise than in the case of the Fast Kurtogram as well as invulnerability to randomimpulseresponsespresent inthesignal. Thefundamental differencebetweenmethodsliesindifferent valuesbeingoptimized. The Fast Kurtogram utilizes kurtosis of the time signal ltered in different bands, whereas the Protrugram takesadvantageofthe kurtosisof envelope spectraamplitudesasa function ofthecenter frequency.0 2000 4000 6000 8000 10000 120000100200Frequency [Hz]KurtosisCF = 8200 HzFig. 28. Protrugram of the jet engine vibration signal, BW=1500 Hz (slightly more than 1xBPFO), step=100 Hz. The bandwidth was set to cover only therstharmonicduetoits veryhighvalue(1325 Hz).0 500 1000 1500012x 10-3Frequency [Hz]Sq.Amplitude [g]500 1000 150012x 10-3Frequency [Hz]1xBPFO = 1325HzFig.29. Narrowbandenvelopespectrumforindicatedoptimalparameters:(a) Protrugramand(b) FastKurtogram.T.Barszcz,A.Jab!onski/MechanicalSystemsandSignalProcessing25(2011)431451 450As discussed in Section 3.2, such an approach enables successful detection of patterns caused by typical REB faults. Intheirfurtherresearch, theauthorswillinvestigateapossiblerelationshipbetweenthewidthofabaseofahillontheProtrugram andthesignicant demodulation bandwidth (seeFig. 9).References[1] J. Antoni, R.B. Randall, The spectral kurtosis: application to the vibratory surveillance and diagnostics of rotating machines, Mechanical Systems andSignalProcessing20(2) (2006)308331.[2] S.Braun,in:MechanicalSignatureAnalysis: TheoryandApplications,Academic Press,NewYork, 1986.[3] R.F. Dwyer, Detection of non-Gaussian signals by frequency domain kurtosis estimation, in: Proceedings of the International Conference on Acoustic,Speech,andSignalProcessing,Boston, 1983,pp.607610.[4] R.F. 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