Research ArticleShort-Sampled Blind Source Separation of Rotating MachinerySignals Based on Spectrum Correction
Xiangdong Huang Xukang Jin and Haipeng Fu
School of Electronic Information Engineering Tianjin University Tianjin 300072 China
Correspondence should be addressed to Haipeng Fu hpfutjueducn
Received 30 March 2016 Accepted 21 June 2016
Academic Editor Carlo Rainieri
Copyright copy 2016 Xiangdong Huang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Nowadays the existing blind source separation (BSS) algorithms in rotatingmachinery fault diagnosis can hardlymeet the demandof fast response high stability and low complexity simultaneouslyTherefore this paper proposes a spectrum correction based BSSalgorithmThrough the incorporation of FFT spectrum correction a screen procedure (consisting of frequencymerging candidatepattern selection and single-source-component recognition) modified 119896-means based source number estimation and mixingmatrix estimation the proposed BSS algorithm can accurately achieve harmonics sensing on field rotating machinery faults in caseof short-sampled observations Both numerical simulation and practical experiment verify the proposed BSS algorithmrsquos superiorityin the recovery quality stability to insufficient samples and efficiency over the existing ICA-based methods Besides rotatingmachinery fault diagnosis the proposed BSS algorithm also possesses a vast potential in other harmonics-related application fields
1 Introduction
As one of themost common classes ofmechanical equipmentrotating machinery plays a significant role in industrialapplications Meanwhile since it generally operates underharsh working conditions it is likely to suffer from failureswhich may cause the machinery to break down or decreasemachinery service performance such as manufacturing qual-ity and operation safety Nowadays rotating machineriesin modern industry tend to be larger more precise andmore automatic which further increases the difficulty of thepotential faults detection
Blind source separation (BSS) which can recover under-lying sources from observations without the knowledge ofthe mixing system is widely used in machinery fault diag-nosis [1ndash5] speech recognition [6] wireless communication[7] and so on Nowadays BSS techniques applied in themachinery fault diagnosis mainly focus on two aspects (1)removal of interferences and disturbances and (2) parametermodeling and feature detection for mechanical faults
On the one hand as is known rotating components (suchas gears and bears) are the common and key components ofmodern machinery [8] Affected by a lot of field factors (such
as multiple motors that are fixed to the same structure orseveral fault events that happen simultaneously) the signalrecorded from a sensor cannot solely reflect the operatingstate of a specific component Furthermore in industrialapplications these recorded signals are inevitably disruptedby the environment (ambient noise other mechanical sys-tems etc) Hence BSS can act as an effective preprocessingprocedure [9] to remove these interferences from other com-ponents or the disturbances arising from the environmentWu et al [10] proposed a BSS algorithm to remove the inter-ferences of acoustic emission signals from amultiple cylinderdiesel engine In [11] an improvedmorphological componentanalysis (MCA) is proposed to diagnose compound faultsof gearboxes Cui et al [4] put forward a null-space pursuit(NSP) BSS algorithm to diagnose compound faults of rollerbearings
On the other hand due to the effect of several rotor oper-ations at some speeds the signal recorded from a vibrationsensor is mainly composed of multiple periodic harmoniccomponents For different categories of faults the spectra ofthese recorded vibration signals exhibit distinct harmonics-related features For example a vibration signal caused byrotor misalignment is mainly characterized with the 2nd
Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 9564938 10 pageshttpdxdoiorg10115520169564938
2 Shock and Vibration
harmonic component [12] The loosening of the bearingin the bearing block often generates components higherthan 10th harmonic (even up to 20th harmonics) The faultof oil whirl [13] always gives rise to some subharmonicsapproximating half harmonic and so forth Hence BSS isexpected to accurately extract these harmonic features ofindividual sources What is more the model-based faultidentification assumes that there exists a certain model tocharacterize a mechanical structure in which the variationof model parameters can reflect the abnormal behaviors ofthe machinery system [14] As a result BSS can be utilized toidentify the model parameters
Hence a lot of studies of BSS problem have been madein the feature extracting and model identification fields Forexample sparse component analysis based [15] and inde-pendent component analysis (ICA) based [14] BSS methodswere employed to estimate the vibration signalsrsquo modalparameters Following this Zvokelj et al [1] proposed theensemble empirical mode decomposition based multiscaleICA (EEMD-MSICA)method and applied it into the bearingfault detection Li et al [16] proposed the supervised ordertracking bounded component analysis (SOTBCA) based BSSalgorithm for gear fault detection which is suitable fordealing with the situation that the vibration signals do notsatisfy the independent condition
To reduce the loss arising from fault accidents it isurgently demanded in field operations that rotating machin-ery fault analysis should be as fast as possible One possiblesolution is to implement the BSS in a short period of observa-tions
However these existing BSS methods can hardly workwell in case of short-sampled observations For example themainstream BSS method in rotating machinery fault diagno-sis is the ICA [17] A lot of ICA-based methods [18 19] orimproved ICA like second-order ICA [20] nonlinear adap-tive ICA [21] and kernel ICA [22] are applied into the failuredetection and analysis As will be elaborated in this paperICA is likely to fall into nondeterministic solutions whenprovided only short-sampled observations This arises fromthe fact that ICA is based on optimizing a kurtosis-relatedobjective function As a fourth-order cumulant statistic thecalculation of kurtosis needs to consume a large amount ofsamples In fact other statistics-based BSS methods suchas fourth-order-only blind identification (FOOBI) method[23] which is based on constructing high-order tensors alsoexhibit poor performance in short-sampled situations
Hence in this paper we propose a novel blind sourceseparation method which works well in both long observa-tions and short observations Due to the incorporation ofspectrum correction and a phase coherence criterion thisBSS method can accurately extract harmonic features (fre-quency amplitude and phase) of individual sources In caseof short-sampled observations which reduce the frequencyresolution of fast Fourier transform (FFT) spectrum andthus deteriorate the picket-fence effect the proposed BSS canalso estimate harmonic parameters by means of spectrumcorrection Moreover a frequency screening procedure con-sisting of frequencymerging candidate pattern selection andsingle-source-component recognition is able to exclude the
interference between individual harmonics-related compo-nentsTherefore unlike ICAor FOOBImethod the proposedBSS is competent in dealing with case of insufficient samplesIn addition the proposed BSS algorithm does not requirethe a priori source number Both numerical simulation andpractical experiment verify the proposed BSS algorithmrsquossuperiority in efficiency and accuracy over the existing ICA-based methods
2 Blind Source Separation Model
21 Temporal Model Consider 119873 underlying sources and119872 recording sensors Suppose that the structure underinvestigation has a high rigidity and the transmission delaysin the mechanical structure are negligible compared to thesampling period [24] In this case the mixing system can betreated as an instantaneous one which can be modeled as
x (119905) = As (119905) + n (119905) (1)
In (1) s(119905) = [1199041(119905) 1199042(119905) 119904
119872(119905)]119879 is the additive noise vector
and A is the mixing matrix The task of short-sampled BSSis to recover the sources 119904
1(119905) sim 119904
119873(119905) from the observations
1199091(119905) sim 119909
119872(119905) without the knowledge of mixing matrix A in
the small sample number situationAccording to the relative relationship between 119873 and
119872 the BSS problem can be divided into 2 conditions theoverdetermined or determined BSS (119873 le 119872) and theunderdetermined BSS (119873 gt 119872) This paper focuses on theoverdetermined condition
Since the vibration of somemechanical component stemsfrom the rotation of the rotor 119899th source can be formulatedas a combination of individual harmonics that is
119904119899(119905) =
119875119899
sum
119901=1
119888119899119901
cos (2120587119891119899119901119905 + 120579119899119901) (2)
where 119875119899is the number of components and 119888
119899119901 119891119899119901 and 120579
119899119901
are the amplitude frequency and phase parameters of 119901thcomponent of 119899th source respectively
Based on this model this paper aims to develop a BSSalgorithm which consumes a small amount of samples toestimate the mixing matrix A and recover all sources 119904
1(119905) sim
119904119873(119905) Besides it should be emphasized that in industrial
applications the source number 119873 is usually not known inadvance Therefore this paper also addresses the problem ofsource number estimation
22 Harmonics Based BSS Model Combining (1) and (2)we can find that if an observation can be further linked to3 harmonic-related parameters 119888
119899119901 119891119899119901 and 120579
119899119901 then the
matrix A is expected to be estimatedSince a real signal contains two conjugate side spectra we
Then it can be inferred from (9) that the frequency-domain vector X(119891
119901lowast) corresponding to the component 119891
119901lowast
is parallel to a119899 Hence as long as sufficient single-source
components 119891119901lowast are collected every column of the mixing
matrix A can be sequentially determined
23 Difficulty of Short-Sampled BSS Note that (7) is an idealFourier model of the BSS system in which the frequency 119891 isa continuous variableHowever as is known the ideal Fouriertransform is unrealizable since it consumes infinite numbersof samples
In practice the ideal Fourier transform is replaced by a119871-point discrete Fourier transform (DFT) (ldquo119871rdquo refers to thenumber of consumed samples) in which 119891 in (7) only allowsbeing one of 119871 frequencies 119896Δ119891 119896 = 0 1 119871 minus 1 (Δ119891 =
119891119904119871 is the frequency resolution and 119891
119904refers to the system
sampling rate) Thus the DFT spectrum of each observationwill suffer from severe picket-fence effect
In addition it is very likely that the frequency 119891119899119901
of 119899thsource is not exactly the integer times of the DFT frequencyresolution Δ119891 = 119891
119904119871 resulting in the fact that the dirac
function 120575(119891 minus 119891119899119901) in (7) cannot achieve an ideal sampling
resultThis deviation is also reflected in119872 observationsrsquo DFTspectra
119898(119896Δ119891) (119898 = 1 119872) which exhibit the effect of
the spectral leakageWithout loss of generality denote the frequency 119891
119899119901of
119899th source as the summation of integer times and fractionaltime of Δ119891 that is
When the sample length 119871 becomes smaller the DFTfrequency unit Δ119891 = 119891
119904119871 gets larger and thus the DFT
spectrum gets coarser Limited by the picket-fence effect infact the fractional item ldquo120575
119899119901Δ119891rdquo in (10) cannot be directly
obtained from DFT bins and thus the frequency 119891119899119901
has tobe treated as the integer times of Δ119891 (ie
119899119901= 119896119899119901Δ119891)
which corresponds to several peak DFT spectral bins ofthe observations As a result large deviation of frequencyestimation inevitably occurs
Furthermore as (7) shows since an observation containsmultiple components severe interinterferences surely occuramong distinct components when these frequency estimates119899119901
are inaccurate As a result the recovered spectrum of119899(119891) is bound to be greatly different with the ideal spectrum
thereby increasing the BSS difficulty in the case of short-sampled observations
To overcome this difficulty we introduce spectrum cor-rection to solve this problem
3 Spectrum Correction Based BSS
31 Spectrum Correction In this paper we apply the ratio-based spectrum correction method addressed in [25] to 119898th(119898 = 1 119872) observation to overcome the short-sampleddifficulty The spectrum correction consists of the followingsteps
4 Shock and Vibration
(1) Implement Hanning-windowed DFT on the 119871-length observation and acquire its DFT spectrum119883119898(119896) (119896 = 0 1 119871 minus 1)
(2) Collect all the peak indices of 119883119898(119896) For the peak
index 119896119898119901119898
119901119898= 1 119875
119898(119875119898is the peak number
of119883119898(119896)) calculate the amplitude ratio V
max 10038161003816100381610038161003816119883119898 (119896119898119901119898 minus 1)1003816100381610038161003816100381610038161003816100381610038161003816119883119898(119896119898119901119898
+ 1)10038161003816100381610038161003816 (11)
Further a variable 119906119898119901119898
can be obtained as
119906119898119901119898
=(2 minus V
119898119901119898
)
(1 + V119898119901119898
) (12)
(3) Adjust 119906119898119901119898
to estimate the fractional number as
119898119901119898
=
119906119898119901119898
if 10038161003816100381610038161003816119883119898 (119896119898119901119898 + 1)10038161003816100381610038161003816gt10038161003816100381610038161003816119883119898(119896119898119901119898
minus 1)10038161003816100381610038161003816
minus119906119898119901119898
else
(13)
Then the accurate frequency estimate is
119898119901119898
=(119896119898119901119898
+ 119898119901119898
) 119891119904
119871 (14)
(4) Acquire the corrected amplitude estimate 119898119901119898
where ldquoangle(sdot)rdquo is the acquiring angle operator
After spectrum correction 3 harmonic parameter sets119898119901119898
119898119901119898
and 119898119901119898
of 119898th observation (119898 =
1 119872) can be acquiredFurther as (8) and (9) demonstrate for an estimated
frequency 119898119901119898
only when it is included by a single sourcecan it be utilized to estimate a column of the mixing matrixA Hence it is necessary to screen those single-source relatedfrequencies from
119898119901119898
119898 = 1 119872
32 Screening Single-Source Components The proposedscheme of screening single-source components consists of 3stages frequency merging candidate pattern selection andsingle-source-component recognition
321 Frequency Merging Its noteworthy that affected bynoise and interferences even for the same single-sourcecomponent its frequency estimates of all the observationsobtained by spectrum correction still exhibit tiny differencesHence a frequency merging procedure should be imple-mented
If we put all these frequency estimates together and sortthem in an ascending order the aforementioned frequencyestimates of tiny differences tend to converge into a clusterAssuming altogether that 119875 clusters are formed without lossof generality denote 119901th (119901 = 1 119875) cluster as
119901119902 119902 =
1 Γ119901 (Γ119901refers to 119901th clusterrsquos size)Then Γ
119901elements of
this cluster can be merged by their average
119891119901=1
Γ119901
Γ119901
sum
119902=1
119901119902 (16)
322 Candidate Pattern Selection Theoretically in terms ofthe BSS model (1) as long as the mixing matrix A doesnot contain zero elements any source component shouldbe included in all the observations In other words thosecorrected frequencies not contained by all the observationscan be treated as fake components and should be removed
In practice given a small threshold 120576 gt 0 for a mergedfrequency119891
119901 if for each observation index119898 (119898 = 1 119872)
there exists only one peak subscript 119901119898satisfying
119891119901can be regarded as an effective component Accordingly
in combination with (9) a pattern vector z119901relevant to this
componentrsquos 119872 corrected parameter pairs (amplitude andphase) can be selected as a candidate vector to estimate acolumn of the matrix A that is
z119901=
[[[[[[[[[[[[
[
11199011
11989011989511199011
119898119901119898
119890119895119898119901119898
119872119901119872
119890119895119872119901119872
]]]]]]]]]]]]
]
119901 = 1 119875 (18)
After candidate pattern selection the number of mergedfrequencies is reduced from 119875 to 119875
323 Single-Source-Component Criterion In rotationalmachinery fault analysis it is likely that multiple sourcescontain some common harmonic components (ie theoverlapping frequencies) Obviously these frequencies arenot in accordance with (8) and (9) and should not be adoptedto estimate the mixing matrix A Hence these overlappingcomponents are invalid and should be removed from thecandidate frequencies 119891
119901
Assume that among 119875 candidate frequencies 119891119901 only
119875lowast frequencies 119891
119901lowast 119901lowast
= 1 119875lowast are single-source
Shock and Vibration 5
components Since 119891119901lowast only belongs to a single source
in combination with (9) its corresponding single-source-component vector z
119901lowast (ie the item X(119891
119901lowast) in (9)) should be
parallel to a column of the mixing matrix AFurthermore since the matrix A is real-valued from (8)
and (9) one can find that 119872 phases of all the entries ofX(119891119901lowast) originate from the same phase 120579
119899119901of a single sourcersquos
component 119891119901lowast (ie 119891
119899119901in (8)) and thus should be equal to
each otherThus a single-source-component vector z
119901lowast should
exhibit two special properties
(1) Its amplitude vector is parallel to a column of themixing matrix A
(2) Its phase vector possesses a property of coherence inwhich any two phase entries of z
119901lowast should approxi-
mately point to the same direction In other wordsthe following inequality of single-source-componentcriterion should be satisfied
1
1198622119872
sum
119903119897
10038161003816100381610038161003816119903119901lowast minus 119897119901lowast
10038161003816100381610038161003816lt 120585 (19)
where 1 le 119903 119897 le 119872 119903 = 119897 and 120585 is a small positivevalue
33 DB-Index Based Source Number Estimation and 119896-MeansClustering If the source number ldquo119873rdquo is known one candirectly employ a clustering algorithm (such as 119896-meansclustering) on single-source-component vectors z
119901lowast 119901lowast=
1 119875lowast to estimate all the119873 columns of the mixing matrix
A However in industrial applications the source numberldquo119873rdquo is usually unknown in advance Therefore this sectioncombines DB-index [26] with 119896-means clustering to estimate119873 and A
Clearly if the number of clusters is specified as 119868 thenthe conventional 119896-means clustering algorithm can classifyz119901lowast 119901lowast= 1 119875
lowast into 119868 clusters 119862
119894(119894 = 1 119868) whose
entries can be denoted as119862119894= z119894119903119894
119903119894= 1 2 119877
119894 (20)
The relationship between these clusters and the entire set ofsingle-source-component vectors can be expressed as
Davies Bouldin index (DB-index) is used to evaluatethe appropriateness of data partitions [26] of a clusteringalgorithmThe definition of the DB-index is formulated as
119863119868=1
119868
119868
sum
119894=1
max119895
(119866119894+ 119866119895
119872119894119895
) (22)
where 119866119894 119866119895represents the dispersion measurement of two
distinct groups 119862119894 119862119895(assuming their cluster centers are
c119894 c119895) and 119872
119894119895refers to the similarity between these two
groups They are calculated with the following two formulas
Apparently on the one hand the larger119872119894119895is the less the
similarity between 119894th and 119895th clusters is that is the better thepartition discrimination isOn the other hand the smaller thedispersion degree 119866
119894is the higher the concentration degree
of the group 119862119894is As a result the smaller the DB-index
is the more appropriate the data partition is Therefore thesource number estimation can be realized by searching outthe minimum DB-index of the 119896-means algorithm that is
119873 = argmin119868
119863119868 (24)
Once the source number119873 is determined themagnitudeparts of cluster centers c
1 c
119873of groups 119862
1 119862
119873gen-
erated by 119896-means algorithm can be directly treated as thecolumns of the mixing matrix estimate A
34 Summary of the Proposed BSS Recovery Algorithm Hav-ing obtained the overdetermined mixing matrix estimate Athe sources can be recovered by
s (119905) = Aminus1x (119905) (25)
where Aminus1 refers to the pseudoinverse of ATo summarize the proposed BSS algorithm is listed as
follows
Step 1 Implement the procedure of spectrum correctionaddressed in Section 31 on 119909
119898(119905) 119898 = 1 119872 to acquire
the corrected frequency set 119898119901119898
amplitude set 119898119901119898
and phase set
119898119901119898
Step 2 Merge the corrected frequencies 119898119901119898
using (16)Further use (17) and (18) to acquire the candidate vectorsz119901 119901 = 1 119875 Then in terms of the screening criterion
(19) pick out single-source-component vectors z119901lowast 119901lowast=
1 119875lowast from these 119875 candidate patterns
Step 3 Implement the modified 119896-means clustering onsingle-source-component vectors z
119901lowast 119901lowast= 1 119875
lowast to
obtain the final estimate of the source number119873 and mixingmatrix A
Step 4 Calculate the pseudoinverse Aminus1 of A and recover thesource s(119905) by (25)
4 Experiment
In this section both numerical simulation of synthesis signalsand practical mechanical diagnosis experiment are con-ducted to verify the performance of proposed BSS algorithmAs a comparison the results of fast-ICA are also presented
6 Shock and Vibration
Table 1 The recovery correlation coefficients and successful times of numerical experiment
41 Numerical Simulation Consider a 3 times 2 mixing systemexpressed as
A = [[[
059 089
097 047
051 053
]]
]
(26)
Two sources 1199041(119905) and 119904
2(119905) are formulated as
1199041(119905) = cos(2120587100119905 + 120587
9) + 14 cos(2120587200119905 + 2120587
9)
+ 185 cos(2120587400119905 + 1205873)
1199042(119905) = 17 cos(212058750119905 + 120587
36)
+ 08 cos(212058780119905 + 512058718)
+ 12 cos(2120587800119905 + 512058712)
(27)
The sampling rate was fixed as 119891119904= 2000Hz and 4 cases
of sample length (119871 = 400 200 100 70) were taken intoaccount Since fast-ICA needs several iterative operations tooptimize a kurtosis-related objective function which startsfrom a random initialization on the demixing matrix it islikely to fall into failure in case of insufficient samples Hencefor each sample length case 1000 trials were conductedThe times of successful trials of both BSS algorithms wererecorded in Table 1 Moreover among these successful trialscorrelation coefficients between the recovered signals and thesources were statistically averaged and also listed in Table 1Figures 1 and 2 present the recovered results of these twoBSS algorithms in case of long observations (119871 = 400) whileFigures 3 and 4 present the short observation case (119871 = 70)
As Figures 1 and 2 depict both the fast-ICA and proposedalgorithm can acquire high-quality recovered waveforms incase of long observations (119871 = 400 limited by page layoutonly half-duration waveforms are plotted) However whenthe sample length reduces into 119871 = 70 one can observe thatobvious distortions appear in the waveforms recovered byfast-ICA in Figure 3 In contrast there exist no distortionsin the recovered waveforms in Figure 4 reflecting that theproposed BSS algorithm outperforms fast-ICA in dealingwith insufficient samples
s1(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s1(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s2(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s2(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
Figure 1 Numerical results of fast-ICA (119871 = 400)
Table 1 shows that as the sample length decreases thetimes of successful recovery of fast-ICA decline sharply andthe average correlation coefficient also tends to be slightlysmaller accordingly In contrast as Table 1 lists all the trialsof the proposed BSS algorithm for different sample lengthsare successfully conducted and all correlation coefficientsremain 1 This is because unlike fast-ICA the proposed BSSalgorithm is based on spectrum correction related harmonicsanalysis rather than statistical analysis and thus it is insensi-tive to the sample length
42 Mechanical Diagnosis Experiment In this section twopractical fault signals 119904
1(119905) 1199042(119905) collected from field rotating
machineries are treated as sources 1199041(119905) is an imbalance
fault signal with the rotating frequency 896853Hz and 1199042(119905)
is a misalignment fault signal with the rotating frequency1028811Hz The mixing system is the same as the matrixA in (26) Different sample lengths (119871 = 400 200 100 70)were considered In each case 1000 trials were conducted
Figure 3 Numerical results of fast-ICA (119871 = 70)
Figures 5ndash8 present the recovery results of both BSS algo-rithms Table 2 lists their recovery performance indexes
From Figures 5 and 6 one can see that just like therecovery of synthesis signals in (27) both the fast-ICA andthe proposed BSS algorithm can achieve excellent recoveryresults in the long-sample situation (119871 = 400) Neverthelesswhen it comes to the short-sample situation (119871 = 70) the
Figure 5 Practical results of fast-ICA (119871 = 400)
proposed BSS algorithm exhibits better performance than thefast-ICA does
From Table 2 one can see that as the sample lengthdecreases from 119871 = 400 to 70 the proposed BSS algo-rithmrsquos superiority over fast-ICA becomes more obvious Inparticular due to the effect of field noise the correlationcoefficients resulting from the proposed algorithm do notremain 1 but approximate to 1 Hence the proposed BSS
8 Shock and Vibration
Table 2 The recovery correlation coefficients and successful times of numerical experiment
Figure 6 Practical results of the proposed BSS algorithm (119871 = 400)
algorithm outperforms the fast-ICA in rotating machineryfault diagnosis
5 Conclusion
This paper proposes a novel blind source separation algo-rithm based on spectrum correction Both numerical sim-ulation and practical experiment verify the proposed BSSalgorithmrsquos excellent performance In general this algorithmpossesses the following 4 merits
(1) Compared to classical fast-ICA algorithm the pro-posed algorithm can achieve a higher-quality sourcerecovery even in case of short-sample observationsThismeets the demand of fast response of the rotatingmachinery fault analysis
(2) The spectrum correction involved in the proposedalgorithmdoeswell in harmonics information extrac-tion and thus is especially suitable for rotatingmachinery fault analysis As is known most of these
s1(t)
t (s)
1
0
minus1
s1(t)
t (s)
1
0
minus1s2(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
Figure 7 Practical result of fast-ICA (119871 = 70)
faults arise from the rotor malfunction which gener-ates a lot of rotating-frequency related harmonics
(3) The proposed BSS algorithm can accurately deter-mine the underlying source number by means of themodified 119896-means clustering which is in accordancewith practical situation of rotating machinery opera-tions
(4) Unlike fast-ICA the proposedBSS algorithmdoes notinvolve random initialization and iterative operationsand thus possesses a higher stability and lower com-plexity which enhances the reliability and efficiencyof the rotating machinery fault analysis
In fact besides rotating machinery fault analysis har-monics analysis is also frequently encountered in a lot offields such as power harmonics analysis channel estimationin communication radar and sonar Hence the proposedBSS algorithm possesses a vast potential in a wide range ofapplications
Shock and Vibration 9s1(t)
t (s)
1
0
minus1
s1(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
Figure 8 Practical result of proposed BSS algorithm (119871 = 70)
Disclosure
Xiangdong Huang and Haipeng Fu are IEEE members
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under Grant 61271322
References
[1] M Zvokelj S Zupan and I Prebil ldquoEEMD-based multiscaleICAmethod for slewing bearing fault detection and diagnosisrdquoJournal of Sound and Vibration vol 370 pp 394ndash423 2016
[2] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013
[3] Z Li Y Jiang C Hu and Z Peng ldquoRecent progress ondecoupling diagnosis of hybrid failures in gear transmissionsystems using vibration sensor signal a reviewrdquo Measurementvol 90 pp 4ndash19 2016
[4] L Cui C Wu C Ma and H Wang ldquoDiagnosis of rollerbearings compound fault using underdetermined blind sourceseparation algorithm based on null-space pursuitrdquo Shock andVibration vol 2015 Article ID 131489 8 pages 2015
[5] J Chang W Liu H Hu and S Nagarajaiah ldquoImprovedindependent component analysis based modal identification ofhigher damping structuresrdquoMeasurement vol 88 pp 402ndash4162016
[6] G Bao Z Ye X Xu and Y Zhou ldquoA compressed sensingapproach to blind separation of speechmixture based on a two-layer sparsity modelrdquo IEEE Transactions on Audio Speech andLanguage Processing vol 21 no 5 pp 899ndash906 2013
[7] Z-C Sha Z-T Huang Y-Y Zhou and F-H Wang ldquoFre-quency-hopping signals sorting based on underdeterminedblind source separationrdquo IET Communications vol 7 no 14 pp1456ndash1464 2013
[8] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013
[9] C Serviere and P Fabry ldquoBlind source separation of noisyharmonic signals for rotating machine diagnosisrdquo Journal ofSound and Vibration vol 272 no 1-2 pp 317ndash339 2004
[10] W Wu T R Lin and A C C Tan ldquoNormalization and sourceseparation of acoustic emission signals for condition moni-toring and fault detection of multi-cylinder diesel enginesrdquoMechanical Systems and Signal Processing vol 64 article 3837pp 479ndash497 2015
[11] D Yu M Wang and X Cheng ldquoA method for the compoundfault diagnosis of gearboxes based on morphological compo-nent analysisrdquoMeasurement vol 91 pp 519ndash531 2016
[12] A W Lees ldquoMisalignment in rigidly coupled rotorsrdquo Journal ofSound and Vibration vol 305 no 1-2 pp 261ndash271 2007
[13] Y D Chen R Du and L S Qu ldquoFault features of large rotatingmachinery and diagnosis using sensor fusionrdquo Journal of Soundand Vibration vol 188 no 2 pp 227ndash242 1995
[14] Y Yang and S Nagarajaiah ldquoBlind identification of damage intime-varying systems using independent component analysiswith wavelet transformrdquoMechanical Systems and Signal Process-ing vol 47 no 1-2 pp 3ndash20 2014
[15] K Yu K Yang and Y Bai ldquoEstimation of modal parametersusing the sparse component analysis based underdeterminedblind source separationrdquoMechanical Systems and Signal Process-ing vol 45 no 2 pp 302ndash316 2014
[16] Z Li X Yan XWang and Z Peng ldquoDetection of gear cracks ina complex gearbox of wind turbines using supervised boundedcomponent analysis of vibration signals collected from multi-channel sensorsrdquo Journal of Sound and Vibration vol 371 pp406ndash433 2016
[17] A Hyvarinen and E Oja ldquoA fast fixed-point algorithm forindependent component analysisrdquo Neural Computation vol 9no 7 pp 1483ndash1492 1997
[18] S I McNeill and D C Zimmerman ldquoRelating independentcomponents to free-vibration modal responsesrdquo Shock andVibration vol 17 no 2 pp 161ndash170 2010
[19] M J Zuo J Lin and X Fan ldquoFeature separation using ICAfor a one-dimensional time series and its application in faultdetectionrdquo Journal of Sound and Vibration vol 287 no 3 pp614ndash624 2005
[20] A Ypma and P Pajunen ldquoRotating machine vibration anal-ysis with second-order independent component analysisrdquo inProceedings of the 1st International Workshop on IndependentComponent Analysis and Signal Separation vol 99 pp 37ndash421999
[21] M J Roan J G Erling and L H Sibul ldquoA new non-linear adaptive blind source separation approach to gear toothfailure detection and analysisrdquo Mechanical Systems and SignalProcessing vol 16 no 5 pp 719ndash740 2002
10 Shock and Vibration
[22] Z Li X Yan Z Tian C Yuan Z Peng and L Li ldquoBlind vibra-tion component separation and nonlinear feature extractionapplied to the nonstationary vibration signals for the gearboxmulti-fault diagnosisrdquo Measurement Journal of the Interna-tional Measurement Confederation vol 46 no 1 pp 259ndash2712013
[23] L De Lathauwer J Castaing and J-F Cardoso ldquoFourth-ordercumulant-based blind identification of underdetermined mix-turesrdquo IEEE Transactions on Signal Processing vol 55 no 6 part2 pp 2965ndash2973 2007
[24] G Gelle M Colas and C Serviere ldquoBlind source separation atool for rotating machine monitoring by vibrations analysisrdquoJournal of Sound and Vibration vol 248 no 5 pp 865ndash8852001
[25] D Agrez ldquoImproving phase estimation with leakage minimiza-tionrdquo IEEE Transactions on Instrumentation and Measurementvol 54 no 4 pp 1347ndash1353 2005
[26] D L Davies and DW Bouldin ldquoA cluster separation measurerdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 1 no 2 pp 224ndash227 1979
harmonic component [12] The loosening of the bearingin the bearing block often generates components higherthan 10th harmonic (even up to 20th harmonics) The faultof oil whirl [13] always gives rise to some subharmonicsapproximating half harmonic and so forth Hence BSS isexpected to accurately extract these harmonic features ofindividual sources What is more the model-based faultidentification assumes that there exists a certain model tocharacterize a mechanical structure in which the variationof model parameters can reflect the abnormal behaviors ofthe machinery system [14] As a result BSS can be utilized toidentify the model parameters
Hence a lot of studies of BSS problem have been madein the feature extracting and model identification fields Forexample sparse component analysis based [15] and inde-pendent component analysis (ICA) based [14] BSS methodswere employed to estimate the vibration signalsrsquo modalparameters Following this Zvokelj et al [1] proposed theensemble empirical mode decomposition based multiscaleICA (EEMD-MSICA)method and applied it into the bearingfault detection Li et al [16] proposed the supervised ordertracking bounded component analysis (SOTBCA) based BSSalgorithm for gear fault detection which is suitable fordealing with the situation that the vibration signals do notsatisfy the independent condition
To reduce the loss arising from fault accidents it isurgently demanded in field operations that rotating machin-ery fault analysis should be as fast as possible One possiblesolution is to implement the BSS in a short period of observa-tions
However these existing BSS methods can hardly workwell in case of short-sampled observations For example themainstream BSS method in rotating machinery fault diagno-sis is the ICA [17] A lot of ICA-based methods [18 19] orimproved ICA like second-order ICA [20] nonlinear adap-tive ICA [21] and kernel ICA [22] are applied into the failuredetection and analysis As will be elaborated in this paperICA is likely to fall into nondeterministic solutions whenprovided only short-sampled observations This arises fromthe fact that ICA is based on optimizing a kurtosis-relatedobjective function As a fourth-order cumulant statistic thecalculation of kurtosis needs to consume a large amount ofsamples In fact other statistics-based BSS methods suchas fourth-order-only blind identification (FOOBI) method[23] which is based on constructing high-order tensors alsoexhibit poor performance in short-sampled situations
Hence in this paper we propose a novel blind sourceseparation method which works well in both long observa-tions and short observations Due to the incorporation ofspectrum correction and a phase coherence criterion thisBSS method can accurately extract harmonic features (fre-quency amplitude and phase) of individual sources In caseof short-sampled observations which reduce the frequencyresolution of fast Fourier transform (FFT) spectrum andthus deteriorate the picket-fence effect the proposed BSS canalso estimate harmonic parameters by means of spectrumcorrection Moreover a frequency screening procedure con-sisting of frequencymerging candidate pattern selection andsingle-source-component recognition is able to exclude the
interference between individual harmonics-related compo-nentsTherefore unlike ICAor FOOBImethod the proposedBSS is competent in dealing with case of insufficient samplesIn addition the proposed BSS algorithm does not requirethe a priori source number Both numerical simulation andpractical experiment verify the proposed BSS algorithmrsquossuperiority in efficiency and accuracy over the existing ICA-based methods
2 Blind Source Separation Model
21 Temporal Model Consider 119873 underlying sources and119872 recording sensors Suppose that the structure underinvestigation has a high rigidity and the transmission delaysin the mechanical structure are negligible compared to thesampling period [24] In this case the mixing system can betreated as an instantaneous one which can be modeled as
x (119905) = As (119905) + n (119905) (1)
In (1) s(119905) = [1199041(119905) 1199042(119905) 119904
119872(119905)]119879 is the additive noise vector
and A is the mixing matrix The task of short-sampled BSSis to recover the sources 119904
1(119905) sim 119904
119873(119905) from the observations
1199091(119905) sim 119909
119872(119905) without the knowledge of mixing matrix A in
the small sample number situationAccording to the relative relationship between 119873 and
119872 the BSS problem can be divided into 2 conditions theoverdetermined or determined BSS (119873 le 119872) and theunderdetermined BSS (119873 gt 119872) This paper focuses on theoverdetermined condition
Since the vibration of somemechanical component stemsfrom the rotation of the rotor 119899th source can be formulatedas a combination of individual harmonics that is
119904119899(119905) =
119875119899
sum
119901=1
119888119899119901
cos (2120587119891119899119901119905 + 120579119899119901) (2)
where 119875119899is the number of components and 119888
119899119901 119891119899119901 and 120579
119899119901
are the amplitude frequency and phase parameters of 119901thcomponent of 119899th source respectively
Based on this model this paper aims to develop a BSSalgorithm which consumes a small amount of samples toestimate the mixing matrix A and recover all sources 119904
1(119905) sim
119904119873(119905) Besides it should be emphasized that in industrial
applications the source number 119873 is usually not known inadvance Therefore this paper also addresses the problem ofsource number estimation
22 Harmonics Based BSS Model Combining (1) and (2)we can find that if an observation can be further linked to3 harmonic-related parameters 119888
119899119901 119891119899119901 and 120579
119899119901 then the
matrix A is expected to be estimatedSince a real signal contains two conjugate side spectra we
Then it can be inferred from (9) that the frequency-domain vector X(119891
119901lowast) corresponding to the component 119891
119901lowast
is parallel to a119899 Hence as long as sufficient single-source
components 119891119901lowast are collected every column of the mixing
matrix A can be sequentially determined
23 Difficulty of Short-Sampled BSS Note that (7) is an idealFourier model of the BSS system in which the frequency 119891 isa continuous variableHowever as is known the ideal Fouriertransform is unrealizable since it consumes infinite numbersof samples
In practice the ideal Fourier transform is replaced by a119871-point discrete Fourier transform (DFT) (ldquo119871rdquo refers to thenumber of consumed samples) in which 119891 in (7) only allowsbeing one of 119871 frequencies 119896Δ119891 119896 = 0 1 119871 minus 1 (Δ119891 =
119891119904119871 is the frequency resolution and 119891
119904refers to the system
sampling rate) Thus the DFT spectrum of each observationwill suffer from severe picket-fence effect
In addition it is very likely that the frequency 119891119899119901
of 119899thsource is not exactly the integer times of the DFT frequencyresolution Δ119891 = 119891
119904119871 resulting in the fact that the dirac
function 120575(119891 minus 119891119899119901) in (7) cannot achieve an ideal sampling
resultThis deviation is also reflected in119872 observationsrsquo DFTspectra
119898(119896Δ119891) (119898 = 1 119872) which exhibit the effect of
the spectral leakageWithout loss of generality denote the frequency 119891
119899119901of
119899th source as the summation of integer times and fractionaltime of Δ119891 that is
When the sample length 119871 becomes smaller the DFTfrequency unit Δ119891 = 119891
119904119871 gets larger and thus the DFT
spectrum gets coarser Limited by the picket-fence effect infact the fractional item ldquo120575
119899119901Δ119891rdquo in (10) cannot be directly
obtained from DFT bins and thus the frequency 119891119899119901
has tobe treated as the integer times of Δ119891 (ie
119899119901= 119896119899119901Δ119891)
which corresponds to several peak DFT spectral bins ofthe observations As a result large deviation of frequencyestimation inevitably occurs
Furthermore as (7) shows since an observation containsmultiple components severe interinterferences surely occuramong distinct components when these frequency estimates119899119901
are inaccurate As a result the recovered spectrum of119899(119891) is bound to be greatly different with the ideal spectrum
thereby increasing the BSS difficulty in the case of short-sampled observations
To overcome this difficulty we introduce spectrum cor-rection to solve this problem
3 Spectrum Correction Based BSS
31 Spectrum Correction In this paper we apply the ratio-based spectrum correction method addressed in [25] to 119898th(119898 = 1 119872) observation to overcome the short-sampleddifficulty The spectrum correction consists of the followingsteps
4 Shock and Vibration
(1) Implement Hanning-windowed DFT on the 119871-length observation and acquire its DFT spectrum119883119898(119896) (119896 = 0 1 119871 minus 1)
(2) Collect all the peak indices of 119883119898(119896) For the peak
index 119896119898119901119898
119901119898= 1 119875
119898(119875119898is the peak number
of119883119898(119896)) calculate the amplitude ratio V
max 10038161003816100381610038161003816119883119898 (119896119898119901119898 minus 1)1003816100381610038161003816100381610038161003816100381610038161003816119883119898(119896119898119901119898
+ 1)10038161003816100381610038161003816 (11)
Further a variable 119906119898119901119898
can be obtained as
119906119898119901119898
=(2 minus V
119898119901119898
)
(1 + V119898119901119898
) (12)
(3) Adjust 119906119898119901119898
to estimate the fractional number as
119898119901119898
=
119906119898119901119898
if 10038161003816100381610038161003816119883119898 (119896119898119901119898 + 1)10038161003816100381610038161003816gt10038161003816100381610038161003816119883119898(119896119898119901119898
minus 1)10038161003816100381610038161003816
minus119906119898119901119898
else
(13)
Then the accurate frequency estimate is
119898119901119898
=(119896119898119901119898
+ 119898119901119898
) 119891119904
119871 (14)
(4) Acquire the corrected amplitude estimate 119898119901119898
where ldquoangle(sdot)rdquo is the acquiring angle operator
After spectrum correction 3 harmonic parameter sets119898119901119898
119898119901119898
and 119898119901119898
of 119898th observation (119898 =
1 119872) can be acquiredFurther as (8) and (9) demonstrate for an estimated
frequency 119898119901119898
only when it is included by a single sourcecan it be utilized to estimate a column of the mixing matrixA Hence it is necessary to screen those single-source relatedfrequencies from
119898119901119898
119898 = 1 119872
32 Screening Single-Source Components The proposedscheme of screening single-source components consists of 3stages frequency merging candidate pattern selection andsingle-source-component recognition
321 Frequency Merging Its noteworthy that affected bynoise and interferences even for the same single-sourcecomponent its frequency estimates of all the observationsobtained by spectrum correction still exhibit tiny differencesHence a frequency merging procedure should be imple-mented
If we put all these frequency estimates together and sortthem in an ascending order the aforementioned frequencyestimates of tiny differences tend to converge into a clusterAssuming altogether that 119875 clusters are formed without lossof generality denote 119901th (119901 = 1 119875) cluster as
119901119902 119902 =
1 Γ119901 (Γ119901refers to 119901th clusterrsquos size)Then Γ
119901elements of
this cluster can be merged by their average
119891119901=1
Γ119901
Γ119901
sum
119902=1
119901119902 (16)
322 Candidate Pattern Selection Theoretically in terms ofthe BSS model (1) as long as the mixing matrix A doesnot contain zero elements any source component shouldbe included in all the observations In other words thosecorrected frequencies not contained by all the observationscan be treated as fake components and should be removed
In practice given a small threshold 120576 gt 0 for a mergedfrequency119891
119901 if for each observation index119898 (119898 = 1 119872)
there exists only one peak subscript 119901119898satisfying
119891119901can be regarded as an effective component Accordingly
in combination with (9) a pattern vector z119901relevant to this
componentrsquos 119872 corrected parameter pairs (amplitude andphase) can be selected as a candidate vector to estimate acolumn of the matrix A that is
z119901=
[[[[[[[[[[[[
[
11199011
11989011989511199011
119898119901119898
119890119895119898119901119898
119872119901119872
119890119895119872119901119872
]]]]]]]]]]]]
]
119901 = 1 119875 (18)
After candidate pattern selection the number of mergedfrequencies is reduced from 119875 to 119875
323 Single-Source-Component Criterion In rotationalmachinery fault analysis it is likely that multiple sourcescontain some common harmonic components (ie theoverlapping frequencies) Obviously these frequencies arenot in accordance with (8) and (9) and should not be adoptedto estimate the mixing matrix A Hence these overlappingcomponents are invalid and should be removed from thecandidate frequencies 119891
119901
Assume that among 119875 candidate frequencies 119891119901 only
119875lowast frequencies 119891
119901lowast 119901lowast
= 1 119875lowast are single-source
Shock and Vibration 5
components Since 119891119901lowast only belongs to a single source
in combination with (9) its corresponding single-source-component vector z
119901lowast (ie the item X(119891
119901lowast) in (9)) should be
parallel to a column of the mixing matrix AFurthermore since the matrix A is real-valued from (8)
and (9) one can find that 119872 phases of all the entries ofX(119891119901lowast) originate from the same phase 120579
119899119901of a single sourcersquos
component 119891119901lowast (ie 119891
119899119901in (8)) and thus should be equal to
each otherThus a single-source-component vector z
119901lowast should
exhibit two special properties
(1) Its amplitude vector is parallel to a column of themixing matrix A
(2) Its phase vector possesses a property of coherence inwhich any two phase entries of z
119901lowast should approxi-
mately point to the same direction In other wordsthe following inequality of single-source-componentcriterion should be satisfied
1
1198622119872
sum
119903119897
10038161003816100381610038161003816119903119901lowast minus 119897119901lowast
10038161003816100381610038161003816lt 120585 (19)
where 1 le 119903 119897 le 119872 119903 = 119897 and 120585 is a small positivevalue
33 DB-Index Based Source Number Estimation and 119896-MeansClustering If the source number ldquo119873rdquo is known one candirectly employ a clustering algorithm (such as 119896-meansclustering) on single-source-component vectors z
119901lowast 119901lowast=
1 119875lowast to estimate all the119873 columns of the mixing matrix
A However in industrial applications the source numberldquo119873rdquo is usually unknown in advance Therefore this sectioncombines DB-index [26] with 119896-means clustering to estimate119873 and A
Clearly if the number of clusters is specified as 119868 thenthe conventional 119896-means clustering algorithm can classifyz119901lowast 119901lowast= 1 119875
lowast into 119868 clusters 119862
119894(119894 = 1 119868) whose
entries can be denoted as119862119894= z119894119903119894
119903119894= 1 2 119877
119894 (20)
The relationship between these clusters and the entire set ofsingle-source-component vectors can be expressed as
Davies Bouldin index (DB-index) is used to evaluatethe appropriateness of data partitions [26] of a clusteringalgorithmThe definition of the DB-index is formulated as
119863119868=1
119868
119868
sum
119894=1
max119895
(119866119894+ 119866119895
119872119894119895
) (22)
where 119866119894 119866119895represents the dispersion measurement of two
distinct groups 119862119894 119862119895(assuming their cluster centers are
c119894 c119895) and 119872
119894119895refers to the similarity between these two
groups They are calculated with the following two formulas
Apparently on the one hand the larger119872119894119895is the less the
similarity between 119894th and 119895th clusters is that is the better thepartition discrimination isOn the other hand the smaller thedispersion degree 119866
119894is the higher the concentration degree
of the group 119862119894is As a result the smaller the DB-index
is the more appropriate the data partition is Therefore thesource number estimation can be realized by searching outthe minimum DB-index of the 119896-means algorithm that is
119873 = argmin119868
119863119868 (24)
Once the source number119873 is determined themagnitudeparts of cluster centers c
1 c
119873of groups 119862
1 119862
119873gen-
erated by 119896-means algorithm can be directly treated as thecolumns of the mixing matrix estimate A
34 Summary of the Proposed BSS Recovery Algorithm Hav-ing obtained the overdetermined mixing matrix estimate Athe sources can be recovered by
s (119905) = Aminus1x (119905) (25)
where Aminus1 refers to the pseudoinverse of ATo summarize the proposed BSS algorithm is listed as
follows
Step 1 Implement the procedure of spectrum correctionaddressed in Section 31 on 119909
119898(119905) 119898 = 1 119872 to acquire
the corrected frequency set 119898119901119898
amplitude set 119898119901119898
and phase set
119898119901119898
Step 2 Merge the corrected frequencies 119898119901119898
using (16)Further use (17) and (18) to acquire the candidate vectorsz119901 119901 = 1 119875 Then in terms of the screening criterion
(19) pick out single-source-component vectors z119901lowast 119901lowast=
1 119875lowast from these 119875 candidate patterns
Step 3 Implement the modified 119896-means clustering onsingle-source-component vectors z
119901lowast 119901lowast= 1 119875
lowast to
obtain the final estimate of the source number119873 and mixingmatrix A
Step 4 Calculate the pseudoinverse Aminus1 of A and recover thesource s(119905) by (25)
4 Experiment
In this section both numerical simulation of synthesis signalsand practical mechanical diagnosis experiment are con-ducted to verify the performance of proposed BSS algorithmAs a comparison the results of fast-ICA are also presented
6 Shock and Vibration
Table 1 The recovery correlation coefficients and successful times of numerical experiment
41 Numerical Simulation Consider a 3 times 2 mixing systemexpressed as
A = [[[
059 089
097 047
051 053
]]
]
(26)
Two sources 1199041(119905) and 119904
2(119905) are formulated as
1199041(119905) = cos(2120587100119905 + 120587
9) + 14 cos(2120587200119905 + 2120587
9)
+ 185 cos(2120587400119905 + 1205873)
1199042(119905) = 17 cos(212058750119905 + 120587
36)
+ 08 cos(212058780119905 + 512058718)
+ 12 cos(2120587800119905 + 512058712)
(27)
The sampling rate was fixed as 119891119904= 2000Hz and 4 cases
of sample length (119871 = 400 200 100 70) were taken intoaccount Since fast-ICA needs several iterative operations tooptimize a kurtosis-related objective function which startsfrom a random initialization on the demixing matrix it islikely to fall into failure in case of insufficient samples Hencefor each sample length case 1000 trials were conductedThe times of successful trials of both BSS algorithms wererecorded in Table 1 Moreover among these successful trialscorrelation coefficients between the recovered signals and thesources were statistically averaged and also listed in Table 1Figures 1 and 2 present the recovered results of these twoBSS algorithms in case of long observations (119871 = 400) whileFigures 3 and 4 present the short observation case (119871 = 70)
As Figures 1 and 2 depict both the fast-ICA and proposedalgorithm can acquire high-quality recovered waveforms incase of long observations (119871 = 400 limited by page layoutonly half-duration waveforms are plotted) However whenthe sample length reduces into 119871 = 70 one can observe thatobvious distortions appear in the waveforms recovered byfast-ICA in Figure 3 In contrast there exist no distortionsin the recovered waveforms in Figure 4 reflecting that theproposed BSS algorithm outperforms fast-ICA in dealingwith insufficient samples
s1(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s1(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s2(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s2(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
Figure 1 Numerical results of fast-ICA (119871 = 400)
Table 1 shows that as the sample length decreases thetimes of successful recovery of fast-ICA decline sharply andthe average correlation coefficient also tends to be slightlysmaller accordingly In contrast as Table 1 lists all the trialsof the proposed BSS algorithm for different sample lengthsare successfully conducted and all correlation coefficientsremain 1 This is because unlike fast-ICA the proposed BSSalgorithm is based on spectrum correction related harmonicsanalysis rather than statistical analysis and thus it is insensi-tive to the sample length
42 Mechanical Diagnosis Experiment In this section twopractical fault signals 119904
1(119905) 1199042(119905) collected from field rotating
machineries are treated as sources 1199041(119905) is an imbalance
fault signal with the rotating frequency 896853Hz and 1199042(119905)
is a misalignment fault signal with the rotating frequency1028811Hz The mixing system is the same as the matrixA in (26) Different sample lengths (119871 = 400 200 100 70)were considered In each case 1000 trials were conducted
Figure 3 Numerical results of fast-ICA (119871 = 70)
Figures 5ndash8 present the recovery results of both BSS algo-rithms Table 2 lists their recovery performance indexes
From Figures 5 and 6 one can see that just like therecovery of synthesis signals in (27) both the fast-ICA andthe proposed BSS algorithm can achieve excellent recoveryresults in the long-sample situation (119871 = 400) Neverthelesswhen it comes to the short-sample situation (119871 = 70) the
Figure 5 Practical results of fast-ICA (119871 = 400)
proposed BSS algorithm exhibits better performance than thefast-ICA does
From Table 2 one can see that as the sample lengthdecreases from 119871 = 400 to 70 the proposed BSS algo-rithmrsquos superiority over fast-ICA becomes more obvious Inparticular due to the effect of field noise the correlationcoefficients resulting from the proposed algorithm do notremain 1 but approximate to 1 Hence the proposed BSS
8 Shock and Vibration
Table 2 The recovery correlation coefficients and successful times of numerical experiment
Figure 6 Practical results of the proposed BSS algorithm (119871 = 400)
algorithm outperforms the fast-ICA in rotating machineryfault diagnosis
5 Conclusion
This paper proposes a novel blind source separation algo-rithm based on spectrum correction Both numerical sim-ulation and practical experiment verify the proposed BSSalgorithmrsquos excellent performance In general this algorithmpossesses the following 4 merits
(1) Compared to classical fast-ICA algorithm the pro-posed algorithm can achieve a higher-quality sourcerecovery even in case of short-sample observationsThismeets the demand of fast response of the rotatingmachinery fault analysis
(2) The spectrum correction involved in the proposedalgorithmdoeswell in harmonics information extrac-tion and thus is especially suitable for rotatingmachinery fault analysis As is known most of these
s1(t)
t (s)
1
0
minus1
s1(t)
t (s)
1
0
minus1s2(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
Figure 7 Practical result of fast-ICA (119871 = 70)
faults arise from the rotor malfunction which gener-ates a lot of rotating-frequency related harmonics
(3) The proposed BSS algorithm can accurately deter-mine the underlying source number by means of themodified 119896-means clustering which is in accordancewith practical situation of rotating machinery opera-tions
(4) Unlike fast-ICA the proposedBSS algorithmdoes notinvolve random initialization and iterative operationsand thus possesses a higher stability and lower com-plexity which enhances the reliability and efficiencyof the rotating machinery fault analysis
In fact besides rotating machinery fault analysis har-monics analysis is also frequently encountered in a lot offields such as power harmonics analysis channel estimationin communication radar and sonar Hence the proposedBSS algorithm possesses a vast potential in a wide range ofapplications
Shock and Vibration 9s1(t)
t (s)
1
0
minus1
s1(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
Figure 8 Practical result of proposed BSS algorithm (119871 = 70)
Disclosure
Xiangdong Huang and Haipeng Fu are IEEE members
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under Grant 61271322
References
[1] M Zvokelj S Zupan and I Prebil ldquoEEMD-based multiscaleICAmethod for slewing bearing fault detection and diagnosisrdquoJournal of Sound and Vibration vol 370 pp 394ndash423 2016
[2] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013
[3] Z Li Y Jiang C Hu and Z Peng ldquoRecent progress ondecoupling diagnosis of hybrid failures in gear transmissionsystems using vibration sensor signal a reviewrdquo Measurementvol 90 pp 4ndash19 2016
[4] L Cui C Wu C Ma and H Wang ldquoDiagnosis of rollerbearings compound fault using underdetermined blind sourceseparation algorithm based on null-space pursuitrdquo Shock andVibration vol 2015 Article ID 131489 8 pages 2015
[5] J Chang W Liu H Hu and S Nagarajaiah ldquoImprovedindependent component analysis based modal identification ofhigher damping structuresrdquoMeasurement vol 88 pp 402ndash4162016
[6] G Bao Z Ye X Xu and Y Zhou ldquoA compressed sensingapproach to blind separation of speechmixture based on a two-layer sparsity modelrdquo IEEE Transactions on Audio Speech andLanguage Processing vol 21 no 5 pp 899ndash906 2013
[7] Z-C Sha Z-T Huang Y-Y Zhou and F-H Wang ldquoFre-quency-hopping signals sorting based on underdeterminedblind source separationrdquo IET Communications vol 7 no 14 pp1456ndash1464 2013
[8] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013
[9] C Serviere and P Fabry ldquoBlind source separation of noisyharmonic signals for rotating machine diagnosisrdquo Journal ofSound and Vibration vol 272 no 1-2 pp 317ndash339 2004
[10] W Wu T R Lin and A C C Tan ldquoNormalization and sourceseparation of acoustic emission signals for condition moni-toring and fault detection of multi-cylinder diesel enginesrdquoMechanical Systems and Signal Processing vol 64 article 3837pp 479ndash497 2015
[11] D Yu M Wang and X Cheng ldquoA method for the compoundfault diagnosis of gearboxes based on morphological compo-nent analysisrdquoMeasurement vol 91 pp 519ndash531 2016
[12] A W Lees ldquoMisalignment in rigidly coupled rotorsrdquo Journal ofSound and Vibration vol 305 no 1-2 pp 261ndash271 2007
[13] Y D Chen R Du and L S Qu ldquoFault features of large rotatingmachinery and diagnosis using sensor fusionrdquo Journal of Soundand Vibration vol 188 no 2 pp 227ndash242 1995
[14] Y Yang and S Nagarajaiah ldquoBlind identification of damage intime-varying systems using independent component analysiswith wavelet transformrdquoMechanical Systems and Signal Process-ing vol 47 no 1-2 pp 3ndash20 2014
[15] K Yu K Yang and Y Bai ldquoEstimation of modal parametersusing the sparse component analysis based underdeterminedblind source separationrdquoMechanical Systems and Signal Process-ing vol 45 no 2 pp 302ndash316 2014
[16] Z Li X Yan XWang and Z Peng ldquoDetection of gear cracks ina complex gearbox of wind turbines using supervised boundedcomponent analysis of vibration signals collected from multi-channel sensorsrdquo Journal of Sound and Vibration vol 371 pp406ndash433 2016
[17] A Hyvarinen and E Oja ldquoA fast fixed-point algorithm forindependent component analysisrdquo Neural Computation vol 9no 7 pp 1483ndash1492 1997
[18] S I McNeill and D C Zimmerman ldquoRelating independentcomponents to free-vibration modal responsesrdquo Shock andVibration vol 17 no 2 pp 161ndash170 2010
[19] M J Zuo J Lin and X Fan ldquoFeature separation using ICAfor a one-dimensional time series and its application in faultdetectionrdquo Journal of Sound and Vibration vol 287 no 3 pp614ndash624 2005
[20] A Ypma and P Pajunen ldquoRotating machine vibration anal-ysis with second-order independent component analysisrdquo inProceedings of the 1st International Workshop on IndependentComponent Analysis and Signal Separation vol 99 pp 37ndash421999
[21] M J Roan J G Erling and L H Sibul ldquoA new non-linear adaptive blind source separation approach to gear toothfailure detection and analysisrdquo Mechanical Systems and SignalProcessing vol 16 no 5 pp 719ndash740 2002
10 Shock and Vibration
[22] Z Li X Yan Z Tian C Yuan Z Peng and L Li ldquoBlind vibra-tion component separation and nonlinear feature extractionapplied to the nonstationary vibration signals for the gearboxmulti-fault diagnosisrdquo Measurement Journal of the Interna-tional Measurement Confederation vol 46 no 1 pp 259ndash2712013
[23] L De Lathauwer J Castaing and J-F Cardoso ldquoFourth-ordercumulant-based blind identification of underdetermined mix-turesrdquo IEEE Transactions on Signal Processing vol 55 no 6 part2 pp 2965ndash2973 2007
[24] G Gelle M Colas and C Serviere ldquoBlind source separation atool for rotating machine monitoring by vibrations analysisrdquoJournal of Sound and Vibration vol 248 no 5 pp 865ndash8852001
[25] D Agrez ldquoImproving phase estimation with leakage minimiza-tionrdquo IEEE Transactions on Instrumentation and Measurementvol 54 no 4 pp 1347ndash1353 2005
[26] D L Davies and DW Bouldin ldquoA cluster separation measurerdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 1 no 2 pp 224ndash227 1979
Then it can be inferred from (9) that the frequency-domain vector X(119891
119901lowast) corresponding to the component 119891
119901lowast
is parallel to a119899 Hence as long as sufficient single-source
components 119891119901lowast are collected every column of the mixing
matrix A can be sequentially determined
23 Difficulty of Short-Sampled BSS Note that (7) is an idealFourier model of the BSS system in which the frequency 119891 isa continuous variableHowever as is known the ideal Fouriertransform is unrealizable since it consumes infinite numbersof samples
In practice the ideal Fourier transform is replaced by a119871-point discrete Fourier transform (DFT) (ldquo119871rdquo refers to thenumber of consumed samples) in which 119891 in (7) only allowsbeing one of 119871 frequencies 119896Δ119891 119896 = 0 1 119871 minus 1 (Δ119891 =
119891119904119871 is the frequency resolution and 119891
119904refers to the system
sampling rate) Thus the DFT spectrum of each observationwill suffer from severe picket-fence effect
In addition it is very likely that the frequency 119891119899119901
of 119899thsource is not exactly the integer times of the DFT frequencyresolution Δ119891 = 119891
119904119871 resulting in the fact that the dirac
function 120575(119891 minus 119891119899119901) in (7) cannot achieve an ideal sampling
resultThis deviation is also reflected in119872 observationsrsquo DFTspectra
119898(119896Δ119891) (119898 = 1 119872) which exhibit the effect of
the spectral leakageWithout loss of generality denote the frequency 119891
119899119901of
119899th source as the summation of integer times and fractionaltime of Δ119891 that is
When the sample length 119871 becomes smaller the DFTfrequency unit Δ119891 = 119891
119904119871 gets larger and thus the DFT
spectrum gets coarser Limited by the picket-fence effect infact the fractional item ldquo120575
119899119901Δ119891rdquo in (10) cannot be directly
obtained from DFT bins and thus the frequency 119891119899119901
has tobe treated as the integer times of Δ119891 (ie
119899119901= 119896119899119901Δ119891)
which corresponds to several peak DFT spectral bins ofthe observations As a result large deviation of frequencyestimation inevitably occurs
Furthermore as (7) shows since an observation containsmultiple components severe interinterferences surely occuramong distinct components when these frequency estimates119899119901
are inaccurate As a result the recovered spectrum of119899(119891) is bound to be greatly different with the ideal spectrum
thereby increasing the BSS difficulty in the case of short-sampled observations
To overcome this difficulty we introduce spectrum cor-rection to solve this problem
3 Spectrum Correction Based BSS
31 Spectrum Correction In this paper we apply the ratio-based spectrum correction method addressed in [25] to 119898th(119898 = 1 119872) observation to overcome the short-sampleddifficulty The spectrum correction consists of the followingsteps
4 Shock and Vibration
(1) Implement Hanning-windowed DFT on the 119871-length observation and acquire its DFT spectrum119883119898(119896) (119896 = 0 1 119871 minus 1)
(2) Collect all the peak indices of 119883119898(119896) For the peak
index 119896119898119901119898
119901119898= 1 119875
119898(119875119898is the peak number
of119883119898(119896)) calculate the amplitude ratio V
max 10038161003816100381610038161003816119883119898 (119896119898119901119898 minus 1)1003816100381610038161003816100381610038161003816100381610038161003816119883119898(119896119898119901119898
+ 1)10038161003816100381610038161003816 (11)
Further a variable 119906119898119901119898
can be obtained as
119906119898119901119898
=(2 minus V
119898119901119898
)
(1 + V119898119901119898
) (12)
(3) Adjust 119906119898119901119898
to estimate the fractional number as
119898119901119898
=
119906119898119901119898
if 10038161003816100381610038161003816119883119898 (119896119898119901119898 + 1)10038161003816100381610038161003816gt10038161003816100381610038161003816119883119898(119896119898119901119898
minus 1)10038161003816100381610038161003816
minus119906119898119901119898
else
(13)
Then the accurate frequency estimate is
119898119901119898
=(119896119898119901119898
+ 119898119901119898
) 119891119904
119871 (14)
(4) Acquire the corrected amplitude estimate 119898119901119898
where ldquoangle(sdot)rdquo is the acquiring angle operator
After spectrum correction 3 harmonic parameter sets119898119901119898
119898119901119898
and 119898119901119898
of 119898th observation (119898 =
1 119872) can be acquiredFurther as (8) and (9) demonstrate for an estimated
frequency 119898119901119898
only when it is included by a single sourcecan it be utilized to estimate a column of the mixing matrixA Hence it is necessary to screen those single-source relatedfrequencies from
119898119901119898
119898 = 1 119872
32 Screening Single-Source Components The proposedscheme of screening single-source components consists of 3stages frequency merging candidate pattern selection andsingle-source-component recognition
321 Frequency Merging Its noteworthy that affected bynoise and interferences even for the same single-sourcecomponent its frequency estimates of all the observationsobtained by spectrum correction still exhibit tiny differencesHence a frequency merging procedure should be imple-mented
If we put all these frequency estimates together and sortthem in an ascending order the aforementioned frequencyestimates of tiny differences tend to converge into a clusterAssuming altogether that 119875 clusters are formed without lossof generality denote 119901th (119901 = 1 119875) cluster as
119901119902 119902 =
1 Γ119901 (Γ119901refers to 119901th clusterrsquos size)Then Γ
119901elements of
this cluster can be merged by their average
119891119901=1
Γ119901
Γ119901
sum
119902=1
119901119902 (16)
322 Candidate Pattern Selection Theoretically in terms ofthe BSS model (1) as long as the mixing matrix A doesnot contain zero elements any source component shouldbe included in all the observations In other words thosecorrected frequencies not contained by all the observationscan be treated as fake components and should be removed
In practice given a small threshold 120576 gt 0 for a mergedfrequency119891
119901 if for each observation index119898 (119898 = 1 119872)
there exists only one peak subscript 119901119898satisfying
119891119901can be regarded as an effective component Accordingly
in combination with (9) a pattern vector z119901relevant to this
componentrsquos 119872 corrected parameter pairs (amplitude andphase) can be selected as a candidate vector to estimate acolumn of the matrix A that is
z119901=
[[[[[[[[[[[[
[
11199011
11989011989511199011
119898119901119898
119890119895119898119901119898
119872119901119872
119890119895119872119901119872
]]]]]]]]]]]]
]
119901 = 1 119875 (18)
After candidate pattern selection the number of mergedfrequencies is reduced from 119875 to 119875
323 Single-Source-Component Criterion In rotationalmachinery fault analysis it is likely that multiple sourcescontain some common harmonic components (ie theoverlapping frequencies) Obviously these frequencies arenot in accordance with (8) and (9) and should not be adoptedto estimate the mixing matrix A Hence these overlappingcomponents are invalid and should be removed from thecandidate frequencies 119891
119901
Assume that among 119875 candidate frequencies 119891119901 only
119875lowast frequencies 119891
119901lowast 119901lowast
= 1 119875lowast are single-source
Shock and Vibration 5
components Since 119891119901lowast only belongs to a single source
in combination with (9) its corresponding single-source-component vector z
119901lowast (ie the item X(119891
119901lowast) in (9)) should be
parallel to a column of the mixing matrix AFurthermore since the matrix A is real-valued from (8)
and (9) one can find that 119872 phases of all the entries ofX(119891119901lowast) originate from the same phase 120579
119899119901of a single sourcersquos
component 119891119901lowast (ie 119891
119899119901in (8)) and thus should be equal to
each otherThus a single-source-component vector z
119901lowast should
exhibit two special properties
(1) Its amplitude vector is parallel to a column of themixing matrix A
(2) Its phase vector possesses a property of coherence inwhich any two phase entries of z
119901lowast should approxi-
mately point to the same direction In other wordsthe following inequality of single-source-componentcriterion should be satisfied
1
1198622119872
sum
119903119897
10038161003816100381610038161003816119903119901lowast minus 119897119901lowast
10038161003816100381610038161003816lt 120585 (19)
where 1 le 119903 119897 le 119872 119903 = 119897 and 120585 is a small positivevalue
33 DB-Index Based Source Number Estimation and 119896-MeansClustering If the source number ldquo119873rdquo is known one candirectly employ a clustering algorithm (such as 119896-meansclustering) on single-source-component vectors z
119901lowast 119901lowast=
1 119875lowast to estimate all the119873 columns of the mixing matrix
A However in industrial applications the source numberldquo119873rdquo is usually unknown in advance Therefore this sectioncombines DB-index [26] with 119896-means clustering to estimate119873 and A
Clearly if the number of clusters is specified as 119868 thenthe conventional 119896-means clustering algorithm can classifyz119901lowast 119901lowast= 1 119875
lowast into 119868 clusters 119862
119894(119894 = 1 119868) whose
entries can be denoted as119862119894= z119894119903119894
119903119894= 1 2 119877
119894 (20)
The relationship between these clusters and the entire set ofsingle-source-component vectors can be expressed as
Davies Bouldin index (DB-index) is used to evaluatethe appropriateness of data partitions [26] of a clusteringalgorithmThe definition of the DB-index is formulated as
119863119868=1
119868
119868
sum
119894=1
max119895
(119866119894+ 119866119895
119872119894119895
) (22)
where 119866119894 119866119895represents the dispersion measurement of two
distinct groups 119862119894 119862119895(assuming their cluster centers are
c119894 c119895) and 119872
119894119895refers to the similarity between these two
groups They are calculated with the following two formulas
Apparently on the one hand the larger119872119894119895is the less the
similarity between 119894th and 119895th clusters is that is the better thepartition discrimination isOn the other hand the smaller thedispersion degree 119866
119894is the higher the concentration degree
of the group 119862119894is As a result the smaller the DB-index
is the more appropriate the data partition is Therefore thesource number estimation can be realized by searching outthe minimum DB-index of the 119896-means algorithm that is
119873 = argmin119868
119863119868 (24)
Once the source number119873 is determined themagnitudeparts of cluster centers c
1 c
119873of groups 119862
1 119862
119873gen-
erated by 119896-means algorithm can be directly treated as thecolumns of the mixing matrix estimate A
34 Summary of the Proposed BSS Recovery Algorithm Hav-ing obtained the overdetermined mixing matrix estimate Athe sources can be recovered by
s (119905) = Aminus1x (119905) (25)
where Aminus1 refers to the pseudoinverse of ATo summarize the proposed BSS algorithm is listed as
follows
Step 1 Implement the procedure of spectrum correctionaddressed in Section 31 on 119909
119898(119905) 119898 = 1 119872 to acquire
the corrected frequency set 119898119901119898
amplitude set 119898119901119898
and phase set
119898119901119898
Step 2 Merge the corrected frequencies 119898119901119898
using (16)Further use (17) and (18) to acquire the candidate vectorsz119901 119901 = 1 119875 Then in terms of the screening criterion
(19) pick out single-source-component vectors z119901lowast 119901lowast=
1 119875lowast from these 119875 candidate patterns
Step 3 Implement the modified 119896-means clustering onsingle-source-component vectors z
119901lowast 119901lowast= 1 119875
lowast to
obtain the final estimate of the source number119873 and mixingmatrix A
Step 4 Calculate the pseudoinverse Aminus1 of A and recover thesource s(119905) by (25)
4 Experiment
In this section both numerical simulation of synthesis signalsand practical mechanical diagnosis experiment are con-ducted to verify the performance of proposed BSS algorithmAs a comparison the results of fast-ICA are also presented
6 Shock and Vibration
Table 1 The recovery correlation coefficients and successful times of numerical experiment
41 Numerical Simulation Consider a 3 times 2 mixing systemexpressed as
A = [[[
059 089
097 047
051 053
]]
]
(26)
Two sources 1199041(119905) and 119904
2(119905) are formulated as
1199041(119905) = cos(2120587100119905 + 120587
9) + 14 cos(2120587200119905 + 2120587
9)
+ 185 cos(2120587400119905 + 1205873)
1199042(119905) = 17 cos(212058750119905 + 120587
36)
+ 08 cos(212058780119905 + 512058718)
+ 12 cos(2120587800119905 + 512058712)
(27)
The sampling rate was fixed as 119891119904= 2000Hz and 4 cases
of sample length (119871 = 400 200 100 70) were taken intoaccount Since fast-ICA needs several iterative operations tooptimize a kurtosis-related objective function which startsfrom a random initialization on the demixing matrix it islikely to fall into failure in case of insufficient samples Hencefor each sample length case 1000 trials were conductedThe times of successful trials of both BSS algorithms wererecorded in Table 1 Moreover among these successful trialscorrelation coefficients between the recovered signals and thesources were statistically averaged and also listed in Table 1Figures 1 and 2 present the recovered results of these twoBSS algorithms in case of long observations (119871 = 400) whileFigures 3 and 4 present the short observation case (119871 = 70)
As Figures 1 and 2 depict both the fast-ICA and proposedalgorithm can acquire high-quality recovered waveforms incase of long observations (119871 = 400 limited by page layoutonly half-duration waveforms are plotted) However whenthe sample length reduces into 119871 = 70 one can observe thatobvious distortions appear in the waveforms recovered byfast-ICA in Figure 3 In contrast there exist no distortionsin the recovered waveforms in Figure 4 reflecting that theproposed BSS algorithm outperforms fast-ICA in dealingwith insufficient samples
s1(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s1(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s2(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s2(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
Figure 1 Numerical results of fast-ICA (119871 = 400)
Table 1 shows that as the sample length decreases thetimes of successful recovery of fast-ICA decline sharply andthe average correlation coefficient also tends to be slightlysmaller accordingly In contrast as Table 1 lists all the trialsof the proposed BSS algorithm for different sample lengthsare successfully conducted and all correlation coefficientsremain 1 This is because unlike fast-ICA the proposed BSSalgorithm is based on spectrum correction related harmonicsanalysis rather than statistical analysis and thus it is insensi-tive to the sample length
42 Mechanical Diagnosis Experiment In this section twopractical fault signals 119904
1(119905) 1199042(119905) collected from field rotating
machineries are treated as sources 1199041(119905) is an imbalance
fault signal with the rotating frequency 896853Hz and 1199042(119905)
is a misalignment fault signal with the rotating frequency1028811Hz The mixing system is the same as the matrixA in (26) Different sample lengths (119871 = 400 200 100 70)were considered In each case 1000 trials were conducted
Figure 3 Numerical results of fast-ICA (119871 = 70)
Figures 5ndash8 present the recovery results of both BSS algo-rithms Table 2 lists their recovery performance indexes
From Figures 5 and 6 one can see that just like therecovery of synthesis signals in (27) both the fast-ICA andthe proposed BSS algorithm can achieve excellent recoveryresults in the long-sample situation (119871 = 400) Neverthelesswhen it comes to the short-sample situation (119871 = 70) the
Figure 5 Practical results of fast-ICA (119871 = 400)
proposed BSS algorithm exhibits better performance than thefast-ICA does
From Table 2 one can see that as the sample lengthdecreases from 119871 = 400 to 70 the proposed BSS algo-rithmrsquos superiority over fast-ICA becomes more obvious Inparticular due to the effect of field noise the correlationcoefficients resulting from the proposed algorithm do notremain 1 but approximate to 1 Hence the proposed BSS
8 Shock and Vibration
Table 2 The recovery correlation coefficients and successful times of numerical experiment
Figure 6 Practical results of the proposed BSS algorithm (119871 = 400)
algorithm outperforms the fast-ICA in rotating machineryfault diagnosis
5 Conclusion
This paper proposes a novel blind source separation algo-rithm based on spectrum correction Both numerical sim-ulation and practical experiment verify the proposed BSSalgorithmrsquos excellent performance In general this algorithmpossesses the following 4 merits
(1) Compared to classical fast-ICA algorithm the pro-posed algorithm can achieve a higher-quality sourcerecovery even in case of short-sample observationsThismeets the demand of fast response of the rotatingmachinery fault analysis
(2) The spectrum correction involved in the proposedalgorithmdoeswell in harmonics information extrac-tion and thus is especially suitable for rotatingmachinery fault analysis As is known most of these
s1(t)
t (s)
1
0
minus1
s1(t)
t (s)
1
0
minus1s2(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
Figure 7 Practical result of fast-ICA (119871 = 70)
faults arise from the rotor malfunction which gener-ates a lot of rotating-frequency related harmonics
(3) The proposed BSS algorithm can accurately deter-mine the underlying source number by means of themodified 119896-means clustering which is in accordancewith practical situation of rotating machinery opera-tions
(4) Unlike fast-ICA the proposedBSS algorithmdoes notinvolve random initialization and iterative operationsand thus possesses a higher stability and lower com-plexity which enhances the reliability and efficiencyof the rotating machinery fault analysis
In fact besides rotating machinery fault analysis har-monics analysis is also frequently encountered in a lot offields such as power harmonics analysis channel estimationin communication radar and sonar Hence the proposedBSS algorithm possesses a vast potential in a wide range ofapplications
Shock and Vibration 9s1(t)
t (s)
1
0
minus1
s1(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
Figure 8 Practical result of proposed BSS algorithm (119871 = 70)
Disclosure
Xiangdong Huang and Haipeng Fu are IEEE members
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under Grant 61271322
References
[1] M Zvokelj S Zupan and I Prebil ldquoEEMD-based multiscaleICAmethod for slewing bearing fault detection and diagnosisrdquoJournal of Sound and Vibration vol 370 pp 394ndash423 2016
[2] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013
[3] Z Li Y Jiang C Hu and Z Peng ldquoRecent progress ondecoupling diagnosis of hybrid failures in gear transmissionsystems using vibration sensor signal a reviewrdquo Measurementvol 90 pp 4ndash19 2016
[4] L Cui C Wu C Ma and H Wang ldquoDiagnosis of rollerbearings compound fault using underdetermined blind sourceseparation algorithm based on null-space pursuitrdquo Shock andVibration vol 2015 Article ID 131489 8 pages 2015
[5] J Chang W Liu H Hu and S Nagarajaiah ldquoImprovedindependent component analysis based modal identification ofhigher damping structuresrdquoMeasurement vol 88 pp 402ndash4162016
[6] G Bao Z Ye X Xu and Y Zhou ldquoA compressed sensingapproach to blind separation of speechmixture based on a two-layer sparsity modelrdquo IEEE Transactions on Audio Speech andLanguage Processing vol 21 no 5 pp 899ndash906 2013
[7] Z-C Sha Z-T Huang Y-Y Zhou and F-H Wang ldquoFre-quency-hopping signals sorting based on underdeterminedblind source separationrdquo IET Communications vol 7 no 14 pp1456ndash1464 2013
[8] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013
[9] C Serviere and P Fabry ldquoBlind source separation of noisyharmonic signals for rotating machine diagnosisrdquo Journal ofSound and Vibration vol 272 no 1-2 pp 317ndash339 2004
[10] W Wu T R Lin and A C C Tan ldquoNormalization and sourceseparation of acoustic emission signals for condition moni-toring and fault detection of multi-cylinder diesel enginesrdquoMechanical Systems and Signal Processing vol 64 article 3837pp 479ndash497 2015
[11] D Yu M Wang and X Cheng ldquoA method for the compoundfault diagnosis of gearboxes based on morphological compo-nent analysisrdquoMeasurement vol 91 pp 519ndash531 2016
[12] A W Lees ldquoMisalignment in rigidly coupled rotorsrdquo Journal ofSound and Vibration vol 305 no 1-2 pp 261ndash271 2007
[13] Y D Chen R Du and L S Qu ldquoFault features of large rotatingmachinery and diagnosis using sensor fusionrdquo Journal of Soundand Vibration vol 188 no 2 pp 227ndash242 1995
[14] Y Yang and S Nagarajaiah ldquoBlind identification of damage intime-varying systems using independent component analysiswith wavelet transformrdquoMechanical Systems and Signal Process-ing vol 47 no 1-2 pp 3ndash20 2014
[15] K Yu K Yang and Y Bai ldquoEstimation of modal parametersusing the sparse component analysis based underdeterminedblind source separationrdquoMechanical Systems and Signal Process-ing vol 45 no 2 pp 302ndash316 2014
[16] Z Li X Yan XWang and Z Peng ldquoDetection of gear cracks ina complex gearbox of wind turbines using supervised boundedcomponent analysis of vibration signals collected from multi-channel sensorsrdquo Journal of Sound and Vibration vol 371 pp406ndash433 2016
[17] A Hyvarinen and E Oja ldquoA fast fixed-point algorithm forindependent component analysisrdquo Neural Computation vol 9no 7 pp 1483ndash1492 1997
[18] S I McNeill and D C Zimmerman ldquoRelating independentcomponents to free-vibration modal responsesrdquo Shock andVibration vol 17 no 2 pp 161ndash170 2010
[19] M J Zuo J Lin and X Fan ldquoFeature separation using ICAfor a one-dimensional time series and its application in faultdetectionrdquo Journal of Sound and Vibration vol 287 no 3 pp614ndash624 2005
[20] A Ypma and P Pajunen ldquoRotating machine vibration anal-ysis with second-order independent component analysisrdquo inProceedings of the 1st International Workshop on IndependentComponent Analysis and Signal Separation vol 99 pp 37ndash421999
[21] M J Roan J G Erling and L H Sibul ldquoA new non-linear adaptive blind source separation approach to gear toothfailure detection and analysisrdquo Mechanical Systems and SignalProcessing vol 16 no 5 pp 719ndash740 2002
10 Shock and Vibration
[22] Z Li X Yan Z Tian C Yuan Z Peng and L Li ldquoBlind vibra-tion component separation and nonlinear feature extractionapplied to the nonstationary vibration signals for the gearboxmulti-fault diagnosisrdquo Measurement Journal of the Interna-tional Measurement Confederation vol 46 no 1 pp 259ndash2712013
[23] L De Lathauwer J Castaing and J-F Cardoso ldquoFourth-ordercumulant-based blind identification of underdetermined mix-turesrdquo IEEE Transactions on Signal Processing vol 55 no 6 part2 pp 2965ndash2973 2007
[24] G Gelle M Colas and C Serviere ldquoBlind source separation atool for rotating machine monitoring by vibrations analysisrdquoJournal of Sound and Vibration vol 248 no 5 pp 865ndash8852001
[25] D Agrez ldquoImproving phase estimation with leakage minimiza-tionrdquo IEEE Transactions on Instrumentation and Measurementvol 54 no 4 pp 1347ndash1353 2005
[26] D L Davies and DW Bouldin ldquoA cluster separation measurerdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 1 no 2 pp 224ndash227 1979
max 10038161003816100381610038161003816119883119898 (119896119898119901119898 minus 1)1003816100381610038161003816100381610038161003816100381610038161003816119883119898(119896119898119901119898
+ 1)10038161003816100381610038161003816 (11)
Further a variable 119906119898119901119898
can be obtained as
119906119898119901119898
=(2 minus V
119898119901119898
)
(1 + V119898119901119898
) (12)
(3) Adjust 119906119898119901119898
to estimate the fractional number as
119898119901119898
=
119906119898119901119898
if 10038161003816100381610038161003816119883119898 (119896119898119901119898 + 1)10038161003816100381610038161003816gt10038161003816100381610038161003816119883119898(119896119898119901119898
minus 1)10038161003816100381610038161003816
minus119906119898119901119898
else
(13)
Then the accurate frequency estimate is
119898119901119898
=(119896119898119901119898
+ 119898119901119898
) 119891119904
119871 (14)
(4) Acquire the corrected amplitude estimate 119898119901119898
where ldquoangle(sdot)rdquo is the acquiring angle operator
After spectrum correction 3 harmonic parameter sets119898119901119898
119898119901119898
and 119898119901119898
of 119898th observation (119898 =
1 119872) can be acquiredFurther as (8) and (9) demonstrate for an estimated
frequency 119898119901119898
only when it is included by a single sourcecan it be utilized to estimate a column of the mixing matrixA Hence it is necessary to screen those single-source relatedfrequencies from
119898119901119898
119898 = 1 119872
32 Screening Single-Source Components The proposedscheme of screening single-source components consists of 3stages frequency merging candidate pattern selection andsingle-source-component recognition
321 Frequency Merging Its noteworthy that affected bynoise and interferences even for the same single-sourcecomponent its frequency estimates of all the observationsobtained by spectrum correction still exhibit tiny differencesHence a frequency merging procedure should be imple-mented
If we put all these frequency estimates together and sortthem in an ascending order the aforementioned frequencyestimates of tiny differences tend to converge into a clusterAssuming altogether that 119875 clusters are formed without lossof generality denote 119901th (119901 = 1 119875) cluster as
119901119902 119902 =
1 Γ119901 (Γ119901refers to 119901th clusterrsquos size)Then Γ
119901elements of
this cluster can be merged by their average
119891119901=1
Γ119901
Γ119901
sum
119902=1
119901119902 (16)
322 Candidate Pattern Selection Theoretically in terms ofthe BSS model (1) as long as the mixing matrix A doesnot contain zero elements any source component shouldbe included in all the observations In other words thosecorrected frequencies not contained by all the observationscan be treated as fake components and should be removed
In practice given a small threshold 120576 gt 0 for a mergedfrequency119891
119901 if for each observation index119898 (119898 = 1 119872)
there exists only one peak subscript 119901119898satisfying
119891119901can be regarded as an effective component Accordingly
in combination with (9) a pattern vector z119901relevant to this
componentrsquos 119872 corrected parameter pairs (amplitude andphase) can be selected as a candidate vector to estimate acolumn of the matrix A that is
z119901=
[[[[[[[[[[[[
[
11199011
11989011989511199011
119898119901119898
119890119895119898119901119898
119872119901119872
119890119895119872119901119872
]]]]]]]]]]]]
]
119901 = 1 119875 (18)
After candidate pattern selection the number of mergedfrequencies is reduced from 119875 to 119875
323 Single-Source-Component Criterion In rotationalmachinery fault analysis it is likely that multiple sourcescontain some common harmonic components (ie theoverlapping frequencies) Obviously these frequencies arenot in accordance with (8) and (9) and should not be adoptedto estimate the mixing matrix A Hence these overlappingcomponents are invalid and should be removed from thecandidate frequencies 119891
119901
Assume that among 119875 candidate frequencies 119891119901 only
119875lowast frequencies 119891
119901lowast 119901lowast
= 1 119875lowast are single-source
Shock and Vibration 5
components Since 119891119901lowast only belongs to a single source
in combination with (9) its corresponding single-source-component vector z
119901lowast (ie the item X(119891
119901lowast) in (9)) should be
parallel to a column of the mixing matrix AFurthermore since the matrix A is real-valued from (8)
and (9) one can find that 119872 phases of all the entries ofX(119891119901lowast) originate from the same phase 120579
119899119901of a single sourcersquos
component 119891119901lowast (ie 119891
119899119901in (8)) and thus should be equal to
each otherThus a single-source-component vector z
119901lowast should
exhibit two special properties
(1) Its amplitude vector is parallel to a column of themixing matrix A
(2) Its phase vector possesses a property of coherence inwhich any two phase entries of z
119901lowast should approxi-
mately point to the same direction In other wordsthe following inequality of single-source-componentcriterion should be satisfied
1
1198622119872
sum
119903119897
10038161003816100381610038161003816119903119901lowast minus 119897119901lowast
10038161003816100381610038161003816lt 120585 (19)
where 1 le 119903 119897 le 119872 119903 = 119897 and 120585 is a small positivevalue
33 DB-Index Based Source Number Estimation and 119896-MeansClustering If the source number ldquo119873rdquo is known one candirectly employ a clustering algorithm (such as 119896-meansclustering) on single-source-component vectors z
119901lowast 119901lowast=
1 119875lowast to estimate all the119873 columns of the mixing matrix
A However in industrial applications the source numberldquo119873rdquo is usually unknown in advance Therefore this sectioncombines DB-index [26] with 119896-means clustering to estimate119873 and A
Clearly if the number of clusters is specified as 119868 thenthe conventional 119896-means clustering algorithm can classifyz119901lowast 119901lowast= 1 119875
lowast into 119868 clusters 119862
119894(119894 = 1 119868) whose
entries can be denoted as119862119894= z119894119903119894
119903119894= 1 2 119877
119894 (20)
The relationship between these clusters and the entire set ofsingle-source-component vectors can be expressed as
Davies Bouldin index (DB-index) is used to evaluatethe appropriateness of data partitions [26] of a clusteringalgorithmThe definition of the DB-index is formulated as
119863119868=1
119868
119868
sum
119894=1
max119895
(119866119894+ 119866119895
119872119894119895
) (22)
where 119866119894 119866119895represents the dispersion measurement of two
distinct groups 119862119894 119862119895(assuming their cluster centers are
c119894 c119895) and 119872
119894119895refers to the similarity between these two
groups They are calculated with the following two formulas
Apparently on the one hand the larger119872119894119895is the less the
similarity between 119894th and 119895th clusters is that is the better thepartition discrimination isOn the other hand the smaller thedispersion degree 119866
119894is the higher the concentration degree
of the group 119862119894is As a result the smaller the DB-index
is the more appropriate the data partition is Therefore thesource number estimation can be realized by searching outthe minimum DB-index of the 119896-means algorithm that is
119873 = argmin119868
119863119868 (24)
Once the source number119873 is determined themagnitudeparts of cluster centers c
1 c
119873of groups 119862
1 119862
119873gen-
erated by 119896-means algorithm can be directly treated as thecolumns of the mixing matrix estimate A
34 Summary of the Proposed BSS Recovery Algorithm Hav-ing obtained the overdetermined mixing matrix estimate Athe sources can be recovered by
s (119905) = Aminus1x (119905) (25)
where Aminus1 refers to the pseudoinverse of ATo summarize the proposed BSS algorithm is listed as
follows
Step 1 Implement the procedure of spectrum correctionaddressed in Section 31 on 119909
119898(119905) 119898 = 1 119872 to acquire
the corrected frequency set 119898119901119898
amplitude set 119898119901119898
and phase set
119898119901119898
Step 2 Merge the corrected frequencies 119898119901119898
using (16)Further use (17) and (18) to acquire the candidate vectorsz119901 119901 = 1 119875 Then in terms of the screening criterion
(19) pick out single-source-component vectors z119901lowast 119901lowast=
1 119875lowast from these 119875 candidate patterns
Step 3 Implement the modified 119896-means clustering onsingle-source-component vectors z
119901lowast 119901lowast= 1 119875
lowast to
obtain the final estimate of the source number119873 and mixingmatrix A
Step 4 Calculate the pseudoinverse Aminus1 of A and recover thesource s(119905) by (25)
4 Experiment
In this section both numerical simulation of synthesis signalsand practical mechanical diagnosis experiment are con-ducted to verify the performance of proposed BSS algorithmAs a comparison the results of fast-ICA are also presented
6 Shock and Vibration
Table 1 The recovery correlation coefficients and successful times of numerical experiment
41 Numerical Simulation Consider a 3 times 2 mixing systemexpressed as
A = [[[
059 089
097 047
051 053
]]
]
(26)
Two sources 1199041(119905) and 119904
2(119905) are formulated as
1199041(119905) = cos(2120587100119905 + 120587
9) + 14 cos(2120587200119905 + 2120587
9)
+ 185 cos(2120587400119905 + 1205873)
1199042(119905) = 17 cos(212058750119905 + 120587
36)
+ 08 cos(212058780119905 + 512058718)
+ 12 cos(2120587800119905 + 512058712)
(27)
The sampling rate was fixed as 119891119904= 2000Hz and 4 cases
of sample length (119871 = 400 200 100 70) were taken intoaccount Since fast-ICA needs several iterative operations tooptimize a kurtosis-related objective function which startsfrom a random initialization on the demixing matrix it islikely to fall into failure in case of insufficient samples Hencefor each sample length case 1000 trials were conductedThe times of successful trials of both BSS algorithms wererecorded in Table 1 Moreover among these successful trialscorrelation coefficients between the recovered signals and thesources were statistically averaged and also listed in Table 1Figures 1 and 2 present the recovered results of these twoBSS algorithms in case of long observations (119871 = 400) whileFigures 3 and 4 present the short observation case (119871 = 70)
As Figures 1 and 2 depict both the fast-ICA and proposedalgorithm can acquire high-quality recovered waveforms incase of long observations (119871 = 400 limited by page layoutonly half-duration waveforms are plotted) However whenthe sample length reduces into 119871 = 70 one can observe thatobvious distortions appear in the waveforms recovered byfast-ICA in Figure 3 In contrast there exist no distortionsin the recovered waveforms in Figure 4 reflecting that theproposed BSS algorithm outperforms fast-ICA in dealingwith insufficient samples
s1(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s1(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s2(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s2(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
Figure 1 Numerical results of fast-ICA (119871 = 400)
Table 1 shows that as the sample length decreases thetimes of successful recovery of fast-ICA decline sharply andthe average correlation coefficient also tends to be slightlysmaller accordingly In contrast as Table 1 lists all the trialsof the proposed BSS algorithm for different sample lengthsare successfully conducted and all correlation coefficientsremain 1 This is because unlike fast-ICA the proposed BSSalgorithm is based on spectrum correction related harmonicsanalysis rather than statistical analysis and thus it is insensi-tive to the sample length
42 Mechanical Diagnosis Experiment In this section twopractical fault signals 119904
1(119905) 1199042(119905) collected from field rotating
machineries are treated as sources 1199041(119905) is an imbalance
fault signal with the rotating frequency 896853Hz and 1199042(119905)
is a misalignment fault signal with the rotating frequency1028811Hz The mixing system is the same as the matrixA in (26) Different sample lengths (119871 = 400 200 100 70)were considered In each case 1000 trials were conducted
Figure 3 Numerical results of fast-ICA (119871 = 70)
Figures 5ndash8 present the recovery results of both BSS algo-rithms Table 2 lists their recovery performance indexes
From Figures 5 and 6 one can see that just like therecovery of synthesis signals in (27) both the fast-ICA andthe proposed BSS algorithm can achieve excellent recoveryresults in the long-sample situation (119871 = 400) Neverthelesswhen it comes to the short-sample situation (119871 = 70) the
Figure 5 Practical results of fast-ICA (119871 = 400)
proposed BSS algorithm exhibits better performance than thefast-ICA does
From Table 2 one can see that as the sample lengthdecreases from 119871 = 400 to 70 the proposed BSS algo-rithmrsquos superiority over fast-ICA becomes more obvious Inparticular due to the effect of field noise the correlationcoefficients resulting from the proposed algorithm do notremain 1 but approximate to 1 Hence the proposed BSS
8 Shock and Vibration
Table 2 The recovery correlation coefficients and successful times of numerical experiment
Figure 6 Practical results of the proposed BSS algorithm (119871 = 400)
algorithm outperforms the fast-ICA in rotating machineryfault diagnosis
5 Conclusion
This paper proposes a novel blind source separation algo-rithm based on spectrum correction Both numerical sim-ulation and practical experiment verify the proposed BSSalgorithmrsquos excellent performance In general this algorithmpossesses the following 4 merits
(1) Compared to classical fast-ICA algorithm the pro-posed algorithm can achieve a higher-quality sourcerecovery even in case of short-sample observationsThismeets the demand of fast response of the rotatingmachinery fault analysis
(2) The spectrum correction involved in the proposedalgorithmdoeswell in harmonics information extrac-tion and thus is especially suitable for rotatingmachinery fault analysis As is known most of these
s1(t)
t (s)
1
0
minus1
s1(t)
t (s)
1
0
minus1s2(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
Figure 7 Practical result of fast-ICA (119871 = 70)
faults arise from the rotor malfunction which gener-ates a lot of rotating-frequency related harmonics
(3) The proposed BSS algorithm can accurately deter-mine the underlying source number by means of themodified 119896-means clustering which is in accordancewith practical situation of rotating machinery opera-tions
(4) Unlike fast-ICA the proposedBSS algorithmdoes notinvolve random initialization and iterative operationsand thus possesses a higher stability and lower com-plexity which enhances the reliability and efficiencyof the rotating machinery fault analysis
In fact besides rotating machinery fault analysis har-monics analysis is also frequently encountered in a lot offields such as power harmonics analysis channel estimationin communication radar and sonar Hence the proposedBSS algorithm possesses a vast potential in a wide range ofapplications
Shock and Vibration 9s1(t)
t (s)
1
0
minus1
s1(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
Figure 8 Practical result of proposed BSS algorithm (119871 = 70)
Disclosure
Xiangdong Huang and Haipeng Fu are IEEE members
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under Grant 61271322
References
[1] M Zvokelj S Zupan and I Prebil ldquoEEMD-based multiscaleICAmethod for slewing bearing fault detection and diagnosisrdquoJournal of Sound and Vibration vol 370 pp 394ndash423 2016
[2] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013
[3] Z Li Y Jiang C Hu and Z Peng ldquoRecent progress ondecoupling diagnosis of hybrid failures in gear transmissionsystems using vibration sensor signal a reviewrdquo Measurementvol 90 pp 4ndash19 2016
[4] L Cui C Wu C Ma and H Wang ldquoDiagnosis of rollerbearings compound fault using underdetermined blind sourceseparation algorithm based on null-space pursuitrdquo Shock andVibration vol 2015 Article ID 131489 8 pages 2015
[5] J Chang W Liu H Hu and S Nagarajaiah ldquoImprovedindependent component analysis based modal identification ofhigher damping structuresrdquoMeasurement vol 88 pp 402ndash4162016
[6] G Bao Z Ye X Xu and Y Zhou ldquoA compressed sensingapproach to blind separation of speechmixture based on a two-layer sparsity modelrdquo IEEE Transactions on Audio Speech andLanguage Processing vol 21 no 5 pp 899ndash906 2013
[7] Z-C Sha Z-T Huang Y-Y Zhou and F-H Wang ldquoFre-quency-hopping signals sorting based on underdeterminedblind source separationrdquo IET Communications vol 7 no 14 pp1456ndash1464 2013
[8] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013
[9] C Serviere and P Fabry ldquoBlind source separation of noisyharmonic signals for rotating machine diagnosisrdquo Journal ofSound and Vibration vol 272 no 1-2 pp 317ndash339 2004
[10] W Wu T R Lin and A C C Tan ldquoNormalization and sourceseparation of acoustic emission signals for condition moni-toring and fault detection of multi-cylinder diesel enginesrdquoMechanical Systems and Signal Processing vol 64 article 3837pp 479ndash497 2015
[11] D Yu M Wang and X Cheng ldquoA method for the compoundfault diagnosis of gearboxes based on morphological compo-nent analysisrdquoMeasurement vol 91 pp 519ndash531 2016
[12] A W Lees ldquoMisalignment in rigidly coupled rotorsrdquo Journal ofSound and Vibration vol 305 no 1-2 pp 261ndash271 2007
[13] Y D Chen R Du and L S Qu ldquoFault features of large rotatingmachinery and diagnosis using sensor fusionrdquo Journal of Soundand Vibration vol 188 no 2 pp 227ndash242 1995
[14] Y Yang and S Nagarajaiah ldquoBlind identification of damage intime-varying systems using independent component analysiswith wavelet transformrdquoMechanical Systems and Signal Process-ing vol 47 no 1-2 pp 3ndash20 2014
[15] K Yu K Yang and Y Bai ldquoEstimation of modal parametersusing the sparse component analysis based underdeterminedblind source separationrdquoMechanical Systems and Signal Process-ing vol 45 no 2 pp 302ndash316 2014
[16] Z Li X Yan XWang and Z Peng ldquoDetection of gear cracks ina complex gearbox of wind turbines using supervised boundedcomponent analysis of vibration signals collected from multi-channel sensorsrdquo Journal of Sound and Vibration vol 371 pp406ndash433 2016
[17] A Hyvarinen and E Oja ldquoA fast fixed-point algorithm forindependent component analysisrdquo Neural Computation vol 9no 7 pp 1483ndash1492 1997
[18] S I McNeill and D C Zimmerman ldquoRelating independentcomponents to free-vibration modal responsesrdquo Shock andVibration vol 17 no 2 pp 161ndash170 2010
[19] M J Zuo J Lin and X Fan ldquoFeature separation using ICAfor a one-dimensional time series and its application in faultdetectionrdquo Journal of Sound and Vibration vol 287 no 3 pp614ndash624 2005
[20] A Ypma and P Pajunen ldquoRotating machine vibration anal-ysis with second-order independent component analysisrdquo inProceedings of the 1st International Workshop on IndependentComponent Analysis and Signal Separation vol 99 pp 37ndash421999
[21] M J Roan J G Erling and L H Sibul ldquoA new non-linear adaptive blind source separation approach to gear toothfailure detection and analysisrdquo Mechanical Systems and SignalProcessing vol 16 no 5 pp 719ndash740 2002
10 Shock and Vibration
[22] Z Li X Yan Z Tian C Yuan Z Peng and L Li ldquoBlind vibra-tion component separation and nonlinear feature extractionapplied to the nonstationary vibration signals for the gearboxmulti-fault diagnosisrdquo Measurement Journal of the Interna-tional Measurement Confederation vol 46 no 1 pp 259ndash2712013
[23] L De Lathauwer J Castaing and J-F Cardoso ldquoFourth-ordercumulant-based blind identification of underdetermined mix-turesrdquo IEEE Transactions on Signal Processing vol 55 no 6 part2 pp 2965ndash2973 2007
[24] G Gelle M Colas and C Serviere ldquoBlind source separation atool for rotating machine monitoring by vibrations analysisrdquoJournal of Sound and Vibration vol 248 no 5 pp 865ndash8852001
[25] D Agrez ldquoImproving phase estimation with leakage minimiza-tionrdquo IEEE Transactions on Instrumentation and Measurementvol 54 no 4 pp 1347ndash1353 2005
[26] D L Davies and DW Bouldin ldquoA cluster separation measurerdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 1 no 2 pp 224ndash227 1979
components Since 119891119901lowast only belongs to a single source
in combination with (9) its corresponding single-source-component vector z
119901lowast (ie the item X(119891
119901lowast) in (9)) should be
parallel to a column of the mixing matrix AFurthermore since the matrix A is real-valued from (8)
and (9) one can find that 119872 phases of all the entries ofX(119891119901lowast) originate from the same phase 120579
119899119901of a single sourcersquos
component 119891119901lowast (ie 119891
119899119901in (8)) and thus should be equal to
each otherThus a single-source-component vector z
119901lowast should
exhibit two special properties
(1) Its amplitude vector is parallel to a column of themixing matrix A
(2) Its phase vector possesses a property of coherence inwhich any two phase entries of z
119901lowast should approxi-
mately point to the same direction In other wordsthe following inequality of single-source-componentcriterion should be satisfied
1
1198622119872
sum
119903119897
10038161003816100381610038161003816119903119901lowast minus 119897119901lowast
10038161003816100381610038161003816lt 120585 (19)
where 1 le 119903 119897 le 119872 119903 = 119897 and 120585 is a small positivevalue
33 DB-Index Based Source Number Estimation and 119896-MeansClustering If the source number ldquo119873rdquo is known one candirectly employ a clustering algorithm (such as 119896-meansclustering) on single-source-component vectors z
119901lowast 119901lowast=
1 119875lowast to estimate all the119873 columns of the mixing matrix
A However in industrial applications the source numberldquo119873rdquo is usually unknown in advance Therefore this sectioncombines DB-index [26] with 119896-means clustering to estimate119873 and A
Clearly if the number of clusters is specified as 119868 thenthe conventional 119896-means clustering algorithm can classifyz119901lowast 119901lowast= 1 119875
lowast into 119868 clusters 119862
119894(119894 = 1 119868) whose
entries can be denoted as119862119894= z119894119903119894
119903119894= 1 2 119877
119894 (20)
The relationship between these clusters and the entire set ofsingle-source-component vectors can be expressed as
Davies Bouldin index (DB-index) is used to evaluatethe appropriateness of data partitions [26] of a clusteringalgorithmThe definition of the DB-index is formulated as
119863119868=1
119868
119868
sum
119894=1
max119895
(119866119894+ 119866119895
119872119894119895
) (22)
where 119866119894 119866119895represents the dispersion measurement of two
distinct groups 119862119894 119862119895(assuming their cluster centers are
c119894 c119895) and 119872
119894119895refers to the similarity between these two
groups They are calculated with the following two formulas
Apparently on the one hand the larger119872119894119895is the less the
similarity between 119894th and 119895th clusters is that is the better thepartition discrimination isOn the other hand the smaller thedispersion degree 119866
119894is the higher the concentration degree
of the group 119862119894is As a result the smaller the DB-index
is the more appropriate the data partition is Therefore thesource number estimation can be realized by searching outthe minimum DB-index of the 119896-means algorithm that is
119873 = argmin119868
119863119868 (24)
Once the source number119873 is determined themagnitudeparts of cluster centers c
1 c
119873of groups 119862
1 119862
119873gen-
erated by 119896-means algorithm can be directly treated as thecolumns of the mixing matrix estimate A
34 Summary of the Proposed BSS Recovery Algorithm Hav-ing obtained the overdetermined mixing matrix estimate Athe sources can be recovered by
s (119905) = Aminus1x (119905) (25)
where Aminus1 refers to the pseudoinverse of ATo summarize the proposed BSS algorithm is listed as
follows
Step 1 Implement the procedure of spectrum correctionaddressed in Section 31 on 119909
119898(119905) 119898 = 1 119872 to acquire
the corrected frequency set 119898119901119898
amplitude set 119898119901119898
and phase set
119898119901119898
Step 2 Merge the corrected frequencies 119898119901119898
using (16)Further use (17) and (18) to acquire the candidate vectorsz119901 119901 = 1 119875 Then in terms of the screening criterion
(19) pick out single-source-component vectors z119901lowast 119901lowast=
1 119875lowast from these 119875 candidate patterns
Step 3 Implement the modified 119896-means clustering onsingle-source-component vectors z
119901lowast 119901lowast= 1 119875
lowast to
obtain the final estimate of the source number119873 and mixingmatrix A
Step 4 Calculate the pseudoinverse Aminus1 of A and recover thesource s(119905) by (25)
4 Experiment
In this section both numerical simulation of synthesis signalsand practical mechanical diagnosis experiment are con-ducted to verify the performance of proposed BSS algorithmAs a comparison the results of fast-ICA are also presented
6 Shock and Vibration
Table 1 The recovery correlation coefficients and successful times of numerical experiment
41 Numerical Simulation Consider a 3 times 2 mixing systemexpressed as
A = [[[
059 089
097 047
051 053
]]
]
(26)
Two sources 1199041(119905) and 119904
2(119905) are formulated as
1199041(119905) = cos(2120587100119905 + 120587
9) + 14 cos(2120587200119905 + 2120587
9)
+ 185 cos(2120587400119905 + 1205873)
1199042(119905) = 17 cos(212058750119905 + 120587
36)
+ 08 cos(212058780119905 + 512058718)
+ 12 cos(2120587800119905 + 512058712)
(27)
The sampling rate was fixed as 119891119904= 2000Hz and 4 cases
of sample length (119871 = 400 200 100 70) were taken intoaccount Since fast-ICA needs several iterative operations tooptimize a kurtosis-related objective function which startsfrom a random initialization on the demixing matrix it islikely to fall into failure in case of insufficient samples Hencefor each sample length case 1000 trials were conductedThe times of successful trials of both BSS algorithms wererecorded in Table 1 Moreover among these successful trialscorrelation coefficients between the recovered signals and thesources were statistically averaged and also listed in Table 1Figures 1 and 2 present the recovered results of these twoBSS algorithms in case of long observations (119871 = 400) whileFigures 3 and 4 present the short observation case (119871 = 70)
As Figures 1 and 2 depict both the fast-ICA and proposedalgorithm can acquire high-quality recovered waveforms incase of long observations (119871 = 400 limited by page layoutonly half-duration waveforms are plotted) However whenthe sample length reduces into 119871 = 70 one can observe thatobvious distortions appear in the waveforms recovered byfast-ICA in Figure 3 In contrast there exist no distortionsin the recovered waveforms in Figure 4 reflecting that theproposed BSS algorithm outperforms fast-ICA in dealingwith insufficient samples
s1(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s1(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s2(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s2(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
Figure 1 Numerical results of fast-ICA (119871 = 400)
Table 1 shows that as the sample length decreases thetimes of successful recovery of fast-ICA decline sharply andthe average correlation coefficient also tends to be slightlysmaller accordingly In contrast as Table 1 lists all the trialsof the proposed BSS algorithm for different sample lengthsare successfully conducted and all correlation coefficientsremain 1 This is because unlike fast-ICA the proposed BSSalgorithm is based on spectrum correction related harmonicsanalysis rather than statistical analysis and thus it is insensi-tive to the sample length
42 Mechanical Diagnosis Experiment In this section twopractical fault signals 119904
1(119905) 1199042(119905) collected from field rotating
machineries are treated as sources 1199041(119905) is an imbalance
fault signal with the rotating frequency 896853Hz and 1199042(119905)
is a misalignment fault signal with the rotating frequency1028811Hz The mixing system is the same as the matrixA in (26) Different sample lengths (119871 = 400 200 100 70)were considered In each case 1000 trials were conducted
Figure 3 Numerical results of fast-ICA (119871 = 70)
Figures 5ndash8 present the recovery results of both BSS algo-rithms Table 2 lists their recovery performance indexes
From Figures 5 and 6 one can see that just like therecovery of synthesis signals in (27) both the fast-ICA andthe proposed BSS algorithm can achieve excellent recoveryresults in the long-sample situation (119871 = 400) Neverthelesswhen it comes to the short-sample situation (119871 = 70) the
Figure 5 Practical results of fast-ICA (119871 = 400)
proposed BSS algorithm exhibits better performance than thefast-ICA does
From Table 2 one can see that as the sample lengthdecreases from 119871 = 400 to 70 the proposed BSS algo-rithmrsquos superiority over fast-ICA becomes more obvious Inparticular due to the effect of field noise the correlationcoefficients resulting from the proposed algorithm do notremain 1 but approximate to 1 Hence the proposed BSS
8 Shock and Vibration
Table 2 The recovery correlation coefficients and successful times of numerical experiment
Figure 6 Practical results of the proposed BSS algorithm (119871 = 400)
algorithm outperforms the fast-ICA in rotating machineryfault diagnosis
5 Conclusion
This paper proposes a novel blind source separation algo-rithm based on spectrum correction Both numerical sim-ulation and practical experiment verify the proposed BSSalgorithmrsquos excellent performance In general this algorithmpossesses the following 4 merits
(1) Compared to classical fast-ICA algorithm the pro-posed algorithm can achieve a higher-quality sourcerecovery even in case of short-sample observationsThismeets the demand of fast response of the rotatingmachinery fault analysis
(2) The spectrum correction involved in the proposedalgorithmdoeswell in harmonics information extrac-tion and thus is especially suitable for rotatingmachinery fault analysis As is known most of these
s1(t)
t (s)
1
0
minus1
s1(t)
t (s)
1
0
minus1s2(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
Figure 7 Practical result of fast-ICA (119871 = 70)
faults arise from the rotor malfunction which gener-ates a lot of rotating-frequency related harmonics
(3) The proposed BSS algorithm can accurately deter-mine the underlying source number by means of themodified 119896-means clustering which is in accordancewith practical situation of rotating machinery opera-tions
(4) Unlike fast-ICA the proposedBSS algorithmdoes notinvolve random initialization and iterative operationsand thus possesses a higher stability and lower com-plexity which enhances the reliability and efficiencyof the rotating machinery fault analysis
In fact besides rotating machinery fault analysis har-monics analysis is also frequently encountered in a lot offields such as power harmonics analysis channel estimationin communication radar and sonar Hence the proposedBSS algorithm possesses a vast potential in a wide range ofapplications
Shock and Vibration 9s1(t)
t (s)
1
0
minus1
s1(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
Figure 8 Practical result of proposed BSS algorithm (119871 = 70)
Disclosure
Xiangdong Huang and Haipeng Fu are IEEE members
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under Grant 61271322
References
[1] M Zvokelj S Zupan and I Prebil ldquoEEMD-based multiscaleICAmethod for slewing bearing fault detection and diagnosisrdquoJournal of Sound and Vibration vol 370 pp 394ndash423 2016
[2] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013
[3] Z Li Y Jiang C Hu and Z Peng ldquoRecent progress ondecoupling diagnosis of hybrid failures in gear transmissionsystems using vibration sensor signal a reviewrdquo Measurementvol 90 pp 4ndash19 2016
[4] L Cui C Wu C Ma and H Wang ldquoDiagnosis of rollerbearings compound fault using underdetermined blind sourceseparation algorithm based on null-space pursuitrdquo Shock andVibration vol 2015 Article ID 131489 8 pages 2015
[5] J Chang W Liu H Hu and S Nagarajaiah ldquoImprovedindependent component analysis based modal identification ofhigher damping structuresrdquoMeasurement vol 88 pp 402ndash4162016
[6] G Bao Z Ye X Xu and Y Zhou ldquoA compressed sensingapproach to blind separation of speechmixture based on a two-layer sparsity modelrdquo IEEE Transactions on Audio Speech andLanguage Processing vol 21 no 5 pp 899ndash906 2013
[7] Z-C Sha Z-T Huang Y-Y Zhou and F-H Wang ldquoFre-quency-hopping signals sorting based on underdeterminedblind source separationrdquo IET Communications vol 7 no 14 pp1456ndash1464 2013
[8] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013
[9] C Serviere and P Fabry ldquoBlind source separation of noisyharmonic signals for rotating machine diagnosisrdquo Journal ofSound and Vibration vol 272 no 1-2 pp 317ndash339 2004
[10] W Wu T R Lin and A C C Tan ldquoNormalization and sourceseparation of acoustic emission signals for condition moni-toring and fault detection of multi-cylinder diesel enginesrdquoMechanical Systems and Signal Processing vol 64 article 3837pp 479ndash497 2015
[11] D Yu M Wang and X Cheng ldquoA method for the compoundfault diagnosis of gearboxes based on morphological compo-nent analysisrdquoMeasurement vol 91 pp 519ndash531 2016
[12] A W Lees ldquoMisalignment in rigidly coupled rotorsrdquo Journal ofSound and Vibration vol 305 no 1-2 pp 261ndash271 2007
[13] Y D Chen R Du and L S Qu ldquoFault features of large rotatingmachinery and diagnosis using sensor fusionrdquo Journal of Soundand Vibration vol 188 no 2 pp 227ndash242 1995
[14] Y Yang and S Nagarajaiah ldquoBlind identification of damage intime-varying systems using independent component analysiswith wavelet transformrdquoMechanical Systems and Signal Process-ing vol 47 no 1-2 pp 3ndash20 2014
[15] K Yu K Yang and Y Bai ldquoEstimation of modal parametersusing the sparse component analysis based underdeterminedblind source separationrdquoMechanical Systems and Signal Process-ing vol 45 no 2 pp 302ndash316 2014
[16] Z Li X Yan XWang and Z Peng ldquoDetection of gear cracks ina complex gearbox of wind turbines using supervised boundedcomponent analysis of vibration signals collected from multi-channel sensorsrdquo Journal of Sound and Vibration vol 371 pp406ndash433 2016
[17] A Hyvarinen and E Oja ldquoA fast fixed-point algorithm forindependent component analysisrdquo Neural Computation vol 9no 7 pp 1483ndash1492 1997
[18] S I McNeill and D C Zimmerman ldquoRelating independentcomponents to free-vibration modal responsesrdquo Shock andVibration vol 17 no 2 pp 161ndash170 2010
[19] M J Zuo J Lin and X Fan ldquoFeature separation using ICAfor a one-dimensional time series and its application in faultdetectionrdquo Journal of Sound and Vibration vol 287 no 3 pp614ndash624 2005
[20] A Ypma and P Pajunen ldquoRotating machine vibration anal-ysis with second-order independent component analysisrdquo inProceedings of the 1st International Workshop on IndependentComponent Analysis and Signal Separation vol 99 pp 37ndash421999
[21] M J Roan J G Erling and L H Sibul ldquoA new non-linear adaptive blind source separation approach to gear toothfailure detection and analysisrdquo Mechanical Systems and SignalProcessing vol 16 no 5 pp 719ndash740 2002
10 Shock and Vibration
[22] Z Li X Yan Z Tian C Yuan Z Peng and L Li ldquoBlind vibra-tion component separation and nonlinear feature extractionapplied to the nonstationary vibration signals for the gearboxmulti-fault diagnosisrdquo Measurement Journal of the Interna-tional Measurement Confederation vol 46 no 1 pp 259ndash2712013
[23] L De Lathauwer J Castaing and J-F Cardoso ldquoFourth-ordercumulant-based blind identification of underdetermined mix-turesrdquo IEEE Transactions on Signal Processing vol 55 no 6 part2 pp 2965ndash2973 2007
[24] G Gelle M Colas and C Serviere ldquoBlind source separation atool for rotating machine monitoring by vibrations analysisrdquoJournal of Sound and Vibration vol 248 no 5 pp 865ndash8852001
[25] D Agrez ldquoImproving phase estimation with leakage minimiza-tionrdquo IEEE Transactions on Instrumentation and Measurementvol 54 no 4 pp 1347ndash1353 2005
[26] D L Davies and DW Bouldin ldquoA cluster separation measurerdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 1 no 2 pp 224ndash227 1979
41 Numerical Simulation Consider a 3 times 2 mixing systemexpressed as
A = [[[
059 089
097 047
051 053
]]
]
(26)
Two sources 1199041(119905) and 119904
2(119905) are formulated as
1199041(119905) = cos(2120587100119905 + 120587
9) + 14 cos(2120587200119905 + 2120587
9)
+ 185 cos(2120587400119905 + 1205873)
1199042(119905) = 17 cos(212058750119905 + 120587
36)
+ 08 cos(212058780119905 + 512058718)
+ 12 cos(2120587800119905 + 512058712)
(27)
The sampling rate was fixed as 119891119904= 2000Hz and 4 cases
of sample length (119871 = 400 200 100 70) were taken intoaccount Since fast-ICA needs several iterative operations tooptimize a kurtosis-related objective function which startsfrom a random initialization on the demixing matrix it islikely to fall into failure in case of insufficient samples Hencefor each sample length case 1000 trials were conductedThe times of successful trials of both BSS algorithms wererecorded in Table 1 Moreover among these successful trialscorrelation coefficients between the recovered signals and thesources were statistically averaged and also listed in Table 1Figures 1 and 2 present the recovered results of these twoBSS algorithms in case of long observations (119871 = 400) whileFigures 3 and 4 present the short observation case (119871 = 70)
As Figures 1 and 2 depict both the fast-ICA and proposedalgorithm can acquire high-quality recovered waveforms incase of long observations (119871 = 400 limited by page layoutonly half-duration waveforms are plotted) However whenthe sample length reduces into 119871 = 70 one can observe thatobvious distortions appear in the waveforms recovered byfast-ICA in Figure 3 In contrast there exist no distortionsin the recovered waveforms in Figure 4 reflecting that theproposed BSS algorithm outperforms fast-ICA in dealingwith insufficient samples
s1(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s1(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s2(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
s2(t)
t (s)
1
0
minus1
0 001 002 003 004 005 006 007 008 009 01
Figure 1 Numerical results of fast-ICA (119871 = 400)
Table 1 shows that as the sample length decreases thetimes of successful recovery of fast-ICA decline sharply andthe average correlation coefficient also tends to be slightlysmaller accordingly In contrast as Table 1 lists all the trialsof the proposed BSS algorithm for different sample lengthsare successfully conducted and all correlation coefficientsremain 1 This is because unlike fast-ICA the proposed BSSalgorithm is based on spectrum correction related harmonicsanalysis rather than statistical analysis and thus it is insensi-tive to the sample length
42 Mechanical Diagnosis Experiment In this section twopractical fault signals 119904
1(119905) 1199042(119905) collected from field rotating
machineries are treated as sources 1199041(119905) is an imbalance
fault signal with the rotating frequency 896853Hz and 1199042(119905)
is a misalignment fault signal with the rotating frequency1028811Hz The mixing system is the same as the matrixA in (26) Different sample lengths (119871 = 400 200 100 70)were considered In each case 1000 trials were conducted
Figure 3 Numerical results of fast-ICA (119871 = 70)
Figures 5ndash8 present the recovery results of both BSS algo-rithms Table 2 lists their recovery performance indexes
From Figures 5 and 6 one can see that just like therecovery of synthesis signals in (27) both the fast-ICA andthe proposed BSS algorithm can achieve excellent recoveryresults in the long-sample situation (119871 = 400) Neverthelesswhen it comes to the short-sample situation (119871 = 70) the
Figure 5 Practical results of fast-ICA (119871 = 400)
proposed BSS algorithm exhibits better performance than thefast-ICA does
From Table 2 one can see that as the sample lengthdecreases from 119871 = 400 to 70 the proposed BSS algo-rithmrsquos superiority over fast-ICA becomes more obvious Inparticular due to the effect of field noise the correlationcoefficients resulting from the proposed algorithm do notremain 1 but approximate to 1 Hence the proposed BSS
8 Shock and Vibration
Table 2 The recovery correlation coefficients and successful times of numerical experiment
Figure 6 Practical results of the proposed BSS algorithm (119871 = 400)
algorithm outperforms the fast-ICA in rotating machineryfault diagnosis
5 Conclusion
This paper proposes a novel blind source separation algo-rithm based on spectrum correction Both numerical sim-ulation and practical experiment verify the proposed BSSalgorithmrsquos excellent performance In general this algorithmpossesses the following 4 merits
(1) Compared to classical fast-ICA algorithm the pro-posed algorithm can achieve a higher-quality sourcerecovery even in case of short-sample observationsThismeets the demand of fast response of the rotatingmachinery fault analysis
(2) The spectrum correction involved in the proposedalgorithmdoeswell in harmonics information extrac-tion and thus is especially suitable for rotatingmachinery fault analysis As is known most of these
s1(t)
t (s)
1
0
minus1
s1(t)
t (s)
1
0
minus1s2(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
Figure 7 Practical result of fast-ICA (119871 = 70)
faults arise from the rotor malfunction which gener-ates a lot of rotating-frequency related harmonics
(3) The proposed BSS algorithm can accurately deter-mine the underlying source number by means of themodified 119896-means clustering which is in accordancewith practical situation of rotating machinery opera-tions
(4) Unlike fast-ICA the proposedBSS algorithmdoes notinvolve random initialization and iterative operationsand thus possesses a higher stability and lower com-plexity which enhances the reliability and efficiencyof the rotating machinery fault analysis
In fact besides rotating machinery fault analysis har-monics analysis is also frequently encountered in a lot offields such as power harmonics analysis channel estimationin communication radar and sonar Hence the proposedBSS algorithm possesses a vast potential in a wide range ofapplications
Shock and Vibration 9s1(t)
t (s)
1
0
minus1
s1(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
Figure 8 Practical result of proposed BSS algorithm (119871 = 70)
Disclosure
Xiangdong Huang and Haipeng Fu are IEEE members
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under Grant 61271322
References
[1] M Zvokelj S Zupan and I Prebil ldquoEEMD-based multiscaleICAmethod for slewing bearing fault detection and diagnosisrdquoJournal of Sound and Vibration vol 370 pp 394ndash423 2016
[2] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013
[3] Z Li Y Jiang C Hu and Z Peng ldquoRecent progress ondecoupling diagnosis of hybrid failures in gear transmissionsystems using vibration sensor signal a reviewrdquo Measurementvol 90 pp 4ndash19 2016
[4] L Cui C Wu C Ma and H Wang ldquoDiagnosis of rollerbearings compound fault using underdetermined blind sourceseparation algorithm based on null-space pursuitrdquo Shock andVibration vol 2015 Article ID 131489 8 pages 2015
[5] J Chang W Liu H Hu and S Nagarajaiah ldquoImprovedindependent component analysis based modal identification ofhigher damping structuresrdquoMeasurement vol 88 pp 402ndash4162016
[6] G Bao Z Ye X Xu and Y Zhou ldquoA compressed sensingapproach to blind separation of speechmixture based on a two-layer sparsity modelrdquo IEEE Transactions on Audio Speech andLanguage Processing vol 21 no 5 pp 899ndash906 2013
[7] Z-C Sha Z-T Huang Y-Y Zhou and F-H Wang ldquoFre-quency-hopping signals sorting based on underdeterminedblind source separationrdquo IET Communications vol 7 no 14 pp1456ndash1464 2013
[8] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013
[9] C Serviere and P Fabry ldquoBlind source separation of noisyharmonic signals for rotating machine diagnosisrdquo Journal ofSound and Vibration vol 272 no 1-2 pp 317ndash339 2004
[10] W Wu T R Lin and A C C Tan ldquoNormalization and sourceseparation of acoustic emission signals for condition moni-toring and fault detection of multi-cylinder diesel enginesrdquoMechanical Systems and Signal Processing vol 64 article 3837pp 479ndash497 2015
[11] D Yu M Wang and X Cheng ldquoA method for the compoundfault diagnosis of gearboxes based on morphological compo-nent analysisrdquoMeasurement vol 91 pp 519ndash531 2016
[12] A W Lees ldquoMisalignment in rigidly coupled rotorsrdquo Journal ofSound and Vibration vol 305 no 1-2 pp 261ndash271 2007
[13] Y D Chen R Du and L S Qu ldquoFault features of large rotatingmachinery and diagnosis using sensor fusionrdquo Journal of Soundand Vibration vol 188 no 2 pp 227ndash242 1995
[14] Y Yang and S Nagarajaiah ldquoBlind identification of damage intime-varying systems using independent component analysiswith wavelet transformrdquoMechanical Systems and Signal Process-ing vol 47 no 1-2 pp 3ndash20 2014
[15] K Yu K Yang and Y Bai ldquoEstimation of modal parametersusing the sparse component analysis based underdeterminedblind source separationrdquoMechanical Systems and Signal Process-ing vol 45 no 2 pp 302ndash316 2014
[16] Z Li X Yan XWang and Z Peng ldquoDetection of gear cracks ina complex gearbox of wind turbines using supervised boundedcomponent analysis of vibration signals collected from multi-channel sensorsrdquo Journal of Sound and Vibration vol 371 pp406ndash433 2016
[17] A Hyvarinen and E Oja ldquoA fast fixed-point algorithm forindependent component analysisrdquo Neural Computation vol 9no 7 pp 1483ndash1492 1997
[18] S I McNeill and D C Zimmerman ldquoRelating independentcomponents to free-vibration modal responsesrdquo Shock andVibration vol 17 no 2 pp 161ndash170 2010
[19] M J Zuo J Lin and X Fan ldquoFeature separation using ICAfor a one-dimensional time series and its application in faultdetectionrdquo Journal of Sound and Vibration vol 287 no 3 pp614ndash624 2005
[20] A Ypma and P Pajunen ldquoRotating machine vibration anal-ysis with second-order independent component analysisrdquo inProceedings of the 1st International Workshop on IndependentComponent Analysis and Signal Separation vol 99 pp 37ndash421999
[21] M J Roan J G Erling and L H Sibul ldquoA new non-linear adaptive blind source separation approach to gear toothfailure detection and analysisrdquo Mechanical Systems and SignalProcessing vol 16 no 5 pp 719ndash740 2002
10 Shock and Vibration
[22] Z Li X Yan Z Tian C Yuan Z Peng and L Li ldquoBlind vibra-tion component separation and nonlinear feature extractionapplied to the nonstationary vibration signals for the gearboxmulti-fault diagnosisrdquo Measurement Journal of the Interna-tional Measurement Confederation vol 46 no 1 pp 259ndash2712013
[23] L De Lathauwer J Castaing and J-F Cardoso ldquoFourth-ordercumulant-based blind identification of underdetermined mix-turesrdquo IEEE Transactions on Signal Processing vol 55 no 6 part2 pp 2965ndash2973 2007
[24] G Gelle M Colas and C Serviere ldquoBlind source separation atool for rotating machine monitoring by vibrations analysisrdquoJournal of Sound and Vibration vol 248 no 5 pp 865ndash8852001
[25] D Agrez ldquoImproving phase estimation with leakage minimiza-tionrdquo IEEE Transactions on Instrumentation and Measurementvol 54 no 4 pp 1347ndash1353 2005
[26] D L Davies and DW Bouldin ldquoA cluster separation measurerdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 1 no 2 pp 224ndash227 1979
Figure 3 Numerical results of fast-ICA (119871 = 70)
Figures 5ndash8 present the recovery results of both BSS algo-rithms Table 2 lists their recovery performance indexes
From Figures 5 and 6 one can see that just like therecovery of synthesis signals in (27) both the fast-ICA andthe proposed BSS algorithm can achieve excellent recoveryresults in the long-sample situation (119871 = 400) Neverthelesswhen it comes to the short-sample situation (119871 = 70) the
Figure 5 Practical results of fast-ICA (119871 = 400)
proposed BSS algorithm exhibits better performance than thefast-ICA does
From Table 2 one can see that as the sample lengthdecreases from 119871 = 400 to 70 the proposed BSS algo-rithmrsquos superiority over fast-ICA becomes more obvious Inparticular due to the effect of field noise the correlationcoefficients resulting from the proposed algorithm do notremain 1 but approximate to 1 Hence the proposed BSS
8 Shock and Vibration
Table 2 The recovery correlation coefficients and successful times of numerical experiment
Figure 6 Practical results of the proposed BSS algorithm (119871 = 400)
algorithm outperforms the fast-ICA in rotating machineryfault diagnosis
5 Conclusion
This paper proposes a novel blind source separation algo-rithm based on spectrum correction Both numerical sim-ulation and practical experiment verify the proposed BSSalgorithmrsquos excellent performance In general this algorithmpossesses the following 4 merits
(1) Compared to classical fast-ICA algorithm the pro-posed algorithm can achieve a higher-quality sourcerecovery even in case of short-sample observationsThismeets the demand of fast response of the rotatingmachinery fault analysis
(2) The spectrum correction involved in the proposedalgorithmdoeswell in harmonics information extrac-tion and thus is especially suitable for rotatingmachinery fault analysis As is known most of these
s1(t)
t (s)
1
0
minus1
s1(t)
t (s)
1
0
minus1s2(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
Figure 7 Practical result of fast-ICA (119871 = 70)
faults arise from the rotor malfunction which gener-ates a lot of rotating-frequency related harmonics
(3) The proposed BSS algorithm can accurately deter-mine the underlying source number by means of themodified 119896-means clustering which is in accordancewith practical situation of rotating machinery opera-tions
(4) Unlike fast-ICA the proposedBSS algorithmdoes notinvolve random initialization and iterative operationsand thus possesses a higher stability and lower com-plexity which enhances the reliability and efficiencyof the rotating machinery fault analysis
In fact besides rotating machinery fault analysis har-monics analysis is also frequently encountered in a lot offields such as power harmonics analysis channel estimationin communication radar and sonar Hence the proposedBSS algorithm possesses a vast potential in a wide range ofapplications
Shock and Vibration 9s1(t)
t (s)
1
0
minus1
s1(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
Figure 8 Practical result of proposed BSS algorithm (119871 = 70)
Disclosure
Xiangdong Huang and Haipeng Fu are IEEE members
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under Grant 61271322
References
[1] M Zvokelj S Zupan and I Prebil ldquoEEMD-based multiscaleICAmethod for slewing bearing fault detection and diagnosisrdquoJournal of Sound and Vibration vol 370 pp 394ndash423 2016
[2] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013
[3] Z Li Y Jiang C Hu and Z Peng ldquoRecent progress ondecoupling diagnosis of hybrid failures in gear transmissionsystems using vibration sensor signal a reviewrdquo Measurementvol 90 pp 4ndash19 2016
[4] L Cui C Wu C Ma and H Wang ldquoDiagnosis of rollerbearings compound fault using underdetermined blind sourceseparation algorithm based on null-space pursuitrdquo Shock andVibration vol 2015 Article ID 131489 8 pages 2015
[5] J Chang W Liu H Hu and S Nagarajaiah ldquoImprovedindependent component analysis based modal identification ofhigher damping structuresrdquoMeasurement vol 88 pp 402ndash4162016
[6] G Bao Z Ye X Xu and Y Zhou ldquoA compressed sensingapproach to blind separation of speechmixture based on a two-layer sparsity modelrdquo IEEE Transactions on Audio Speech andLanguage Processing vol 21 no 5 pp 899ndash906 2013
[7] Z-C Sha Z-T Huang Y-Y Zhou and F-H Wang ldquoFre-quency-hopping signals sorting based on underdeterminedblind source separationrdquo IET Communications vol 7 no 14 pp1456ndash1464 2013
[8] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013
[9] C Serviere and P Fabry ldquoBlind source separation of noisyharmonic signals for rotating machine diagnosisrdquo Journal ofSound and Vibration vol 272 no 1-2 pp 317ndash339 2004
[10] W Wu T R Lin and A C C Tan ldquoNormalization and sourceseparation of acoustic emission signals for condition moni-toring and fault detection of multi-cylinder diesel enginesrdquoMechanical Systems and Signal Processing vol 64 article 3837pp 479ndash497 2015
[11] D Yu M Wang and X Cheng ldquoA method for the compoundfault diagnosis of gearboxes based on morphological compo-nent analysisrdquoMeasurement vol 91 pp 519ndash531 2016
[12] A W Lees ldquoMisalignment in rigidly coupled rotorsrdquo Journal ofSound and Vibration vol 305 no 1-2 pp 261ndash271 2007
[13] Y D Chen R Du and L S Qu ldquoFault features of large rotatingmachinery and diagnosis using sensor fusionrdquo Journal of Soundand Vibration vol 188 no 2 pp 227ndash242 1995
[14] Y Yang and S Nagarajaiah ldquoBlind identification of damage intime-varying systems using independent component analysiswith wavelet transformrdquoMechanical Systems and Signal Process-ing vol 47 no 1-2 pp 3ndash20 2014
[15] K Yu K Yang and Y Bai ldquoEstimation of modal parametersusing the sparse component analysis based underdeterminedblind source separationrdquoMechanical Systems and Signal Process-ing vol 45 no 2 pp 302ndash316 2014
[16] Z Li X Yan XWang and Z Peng ldquoDetection of gear cracks ina complex gearbox of wind turbines using supervised boundedcomponent analysis of vibration signals collected from multi-channel sensorsrdquo Journal of Sound and Vibration vol 371 pp406ndash433 2016
[17] A Hyvarinen and E Oja ldquoA fast fixed-point algorithm forindependent component analysisrdquo Neural Computation vol 9no 7 pp 1483ndash1492 1997
[18] S I McNeill and D C Zimmerman ldquoRelating independentcomponents to free-vibration modal responsesrdquo Shock andVibration vol 17 no 2 pp 161ndash170 2010
[19] M J Zuo J Lin and X Fan ldquoFeature separation using ICAfor a one-dimensional time series and its application in faultdetectionrdquo Journal of Sound and Vibration vol 287 no 3 pp614ndash624 2005
[20] A Ypma and P Pajunen ldquoRotating machine vibration anal-ysis with second-order independent component analysisrdquo inProceedings of the 1st International Workshop on IndependentComponent Analysis and Signal Separation vol 99 pp 37ndash421999
[21] M J Roan J G Erling and L H Sibul ldquoA new non-linear adaptive blind source separation approach to gear toothfailure detection and analysisrdquo Mechanical Systems and SignalProcessing vol 16 no 5 pp 719ndash740 2002
10 Shock and Vibration
[22] Z Li X Yan Z Tian C Yuan Z Peng and L Li ldquoBlind vibra-tion component separation and nonlinear feature extractionapplied to the nonstationary vibration signals for the gearboxmulti-fault diagnosisrdquo Measurement Journal of the Interna-tional Measurement Confederation vol 46 no 1 pp 259ndash2712013
[23] L De Lathauwer J Castaing and J-F Cardoso ldquoFourth-ordercumulant-based blind identification of underdetermined mix-turesrdquo IEEE Transactions on Signal Processing vol 55 no 6 part2 pp 2965ndash2973 2007
[24] G Gelle M Colas and C Serviere ldquoBlind source separation atool for rotating machine monitoring by vibrations analysisrdquoJournal of Sound and Vibration vol 248 no 5 pp 865ndash8852001
[25] D Agrez ldquoImproving phase estimation with leakage minimiza-tionrdquo IEEE Transactions on Instrumentation and Measurementvol 54 no 4 pp 1347ndash1353 2005
[26] D L Davies and DW Bouldin ldquoA cluster separation measurerdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 1 no 2 pp 224ndash227 1979
Figure 6 Practical results of the proposed BSS algorithm (119871 = 400)
algorithm outperforms the fast-ICA in rotating machineryfault diagnosis
5 Conclusion
This paper proposes a novel blind source separation algo-rithm based on spectrum correction Both numerical sim-ulation and practical experiment verify the proposed BSSalgorithmrsquos excellent performance In general this algorithmpossesses the following 4 merits
(1) Compared to classical fast-ICA algorithm the pro-posed algorithm can achieve a higher-quality sourcerecovery even in case of short-sample observationsThismeets the demand of fast response of the rotatingmachinery fault analysis
(2) The spectrum correction involved in the proposedalgorithmdoeswell in harmonics information extrac-tion and thus is especially suitable for rotatingmachinery fault analysis As is known most of these
s1(t)
t (s)
1
0
minus1
s1(t)
t (s)
1
0
minus1s2(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
Figure 7 Practical result of fast-ICA (119871 = 70)
faults arise from the rotor malfunction which gener-ates a lot of rotating-frequency related harmonics
(3) The proposed BSS algorithm can accurately deter-mine the underlying source number by means of themodified 119896-means clustering which is in accordancewith practical situation of rotating machinery opera-tions
(4) Unlike fast-ICA the proposedBSS algorithmdoes notinvolve random initialization and iterative operationsand thus possesses a higher stability and lower com-plexity which enhances the reliability and efficiencyof the rotating machinery fault analysis
In fact besides rotating machinery fault analysis har-monics analysis is also frequently encountered in a lot offields such as power harmonics analysis channel estimationin communication radar and sonar Hence the proposedBSS algorithm possesses a vast potential in a wide range ofapplications
Shock and Vibration 9s1(t)
t (s)
1
0
minus1
s1(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
s2(t)
t (s)
1
0
minus1
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
0 0005 001 0015 002 0025 003 0035
Figure 8 Practical result of proposed BSS algorithm (119871 = 70)
Disclosure
Xiangdong Huang and Haipeng Fu are IEEE members
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under Grant 61271322
References
[1] M Zvokelj S Zupan and I Prebil ldquoEEMD-based multiscaleICAmethod for slewing bearing fault detection and diagnosisrdquoJournal of Sound and Vibration vol 370 pp 394ndash423 2016
[2] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013
[3] Z Li Y Jiang C Hu and Z Peng ldquoRecent progress ondecoupling diagnosis of hybrid failures in gear transmissionsystems using vibration sensor signal a reviewrdquo Measurementvol 90 pp 4ndash19 2016
[4] L Cui C Wu C Ma and H Wang ldquoDiagnosis of rollerbearings compound fault using underdetermined blind sourceseparation algorithm based on null-space pursuitrdquo Shock andVibration vol 2015 Article ID 131489 8 pages 2015
[5] J Chang W Liu H Hu and S Nagarajaiah ldquoImprovedindependent component analysis based modal identification ofhigher damping structuresrdquoMeasurement vol 88 pp 402ndash4162016
[6] G Bao Z Ye X Xu and Y Zhou ldquoA compressed sensingapproach to blind separation of speechmixture based on a two-layer sparsity modelrdquo IEEE Transactions on Audio Speech andLanguage Processing vol 21 no 5 pp 899ndash906 2013
[7] Z-C Sha Z-T Huang Y-Y Zhou and F-H Wang ldquoFre-quency-hopping signals sorting based on underdeterminedblind source separationrdquo IET Communications vol 7 no 14 pp1456ndash1464 2013
[8] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013
[9] C Serviere and P Fabry ldquoBlind source separation of noisyharmonic signals for rotating machine diagnosisrdquo Journal ofSound and Vibration vol 272 no 1-2 pp 317ndash339 2004
[10] W Wu T R Lin and A C C Tan ldquoNormalization and sourceseparation of acoustic emission signals for condition moni-toring and fault detection of multi-cylinder diesel enginesrdquoMechanical Systems and Signal Processing vol 64 article 3837pp 479ndash497 2015
[11] D Yu M Wang and X Cheng ldquoA method for the compoundfault diagnosis of gearboxes based on morphological compo-nent analysisrdquoMeasurement vol 91 pp 519ndash531 2016
[12] A W Lees ldquoMisalignment in rigidly coupled rotorsrdquo Journal ofSound and Vibration vol 305 no 1-2 pp 261ndash271 2007
[13] Y D Chen R Du and L S Qu ldquoFault features of large rotatingmachinery and diagnosis using sensor fusionrdquo Journal of Soundand Vibration vol 188 no 2 pp 227ndash242 1995
[14] Y Yang and S Nagarajaiah ldquoBlind identification of damage intime-varying systems using independent component analysiswith wavelet transformrdquoMechanical Systems and Signal Process-ing vol 47 no 1-2 pp 3ndash20 2014
[15] K Yu K Yang and Y Bai ldquoEstimation of modal parametersusing the sparse component analysis based underdeterminedblind source separationrdquoMechanical Systems and Signal Process-ing vol 45 no 2 pp 302ndash316 2014
[16] Z Li X Yan XWang and Z Peng ldquoDetection of gear cracks ina complex gearbox of wind turbines using supervised boundedcomponent analysis of vibration signals collected from multi-channel sensorsrdquo Journal of Sound and Vibration vol 371 pp406ndash433 2016
[17] A Hyvarinen and E Oja ldquoA fast fixed-point algorithm forindependent component analysisrdquo Neural Computation vol 9no 7 pp 1483ndash1492 1997
[18] S I McNeill and D C Zimmerman ldquoRelating independentcomponents to free-vibration modal responsesrdquo Shock andVibration vol 17 no 2 pp 161ndash170 2010
[19] M J Zuo J Lin and X Fan ldquoFeature separation using ICAfor a one-dimensional time series and its application in faultdetectionrdquo Journal of Sound and Vibration vol 287 no 3 pp614ndash624 2005
[20] A Ypma and P Pajunen ldquoRotating machine vibration anal-ysis with second-order independent component analysisrdquo inProceedings of the 1st International Workshop on IndependentComponent Analysis and Signal Separation vol 99 pp 37ndash421999
[21] M J Roan J G Erling and L H Sibul ldquoA new non-linear adaptive blind source separation approach to gear toothfailure detection and analysisrdquo Mechanical Systems and SignalProcessing vol 16 no 5 pp 719ndash740 2002
10 Shock and Vibration
[22] Z Li X Yan Z Tian C Yuan Z Peng and L Li ldquoBlind vibra-tion component separation and nonlinear feature extractionapplied to the nonstationary vibration signals for the gearboxmulti-fault diagnosisrdquo Measurement Journal of the Interna-tional Measurement Confederation vol 46 no 1 pp 259ndash2712013
[23] L De Lathauwer J Castaing and J-F Cardoso ldquoFourth-ordercumulant-based blind identification of underdetermined mix-turesrdquo IEEE Transactions on Signal Processing vol 55 no 6 part2 pp 2965ndash2973 2007
[24] G Gelle M Colas and C Serviere ldquoBlind source separation atool for rotating machine monitoring by vibrations analysisrdquoJournal of Sound and Vibration vol 248 no 5 pp 865ndash8852001
[25] D Agrez ldquoImproving phase estimation with leakage minimiza-tionrdquo IEEE Transactions on Instrumentation and Measurementvol 54 no 4 pp 1347ndash1353 2005
[26] D L Davies and DW Bouldin ldquoA cluster separation measurerdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 1 no 2 pp 224ndash227 1979
Figure 8 Practical result of proposed BSS algorithm (119871 = 70)
Disclosure
Xiangdong Huang and Haipeng Fu are IEEE members
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under Grant 61271322
References
[1] M Zvokelj S Zupan and I Prebil ldquoEEMD-based multiscaleICAmethod for slewing bearing fault detection and diagnosisrdquoJournal of Sound and Vibration vol 370 pp 394ndash423 2016
[2] A Sadhu and B Hazra ldquoA novel damage detection algorithmusing time-series analysis-based blind source separationrdquo Shockand Vibration vol 20 no 3 pp 423ndash438 2013
[3] Z Li Y Jiang C Hu and Z Peng ldquoRecent progress ondecoupling diagnosis of hybrid failures in gear transmissionsystems using vibration sensor signal a reviewrdquo Measurementvol 90 pp 4ndash19 2016
[4] L Cui C Wu C Ma and H Wang ldquoDiagnosis of rollerbearings compound fault using underdetermined blind sourceseparation algorithm based on null-space pursuitrdquo Shock andVibration vol 2015 Article ID 131489 8 pages 2015
[5] J Chang W Liu H Hu and S Nagarajaiah ldquoImprovedindependent component analysis based modal identification ofhigher damping structuresrdquoMeasurement vol 88 pp 402ndash4162016
[6] G Bao Z Ye X Xu and Y Zhou ldquoA compressed sensingapproach to blind separation of speechmixture based on a two-layer sparsity modelrdquo IEEE Transactions on Audio Speech andLanguage Processing vol 21 no 5 pp 899ndash906 2013
[7] Z-C Sha Z-T Huang Y-Y Zhou and F-H Wang ldquoFre-quency-hopping signals sorting based on underdeterminedblind source separationrdquo IET Communications vol 7 no 14 pp1456ndash1464 2013
[8] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013
[9] C Serviere and P Fabry ldquoBlind source separation of noisyharmonic signals for rotating machine diagnosisrdquo Journal ofSound and Vibration vol 272 no 1-2 pp 317ndash339 2004
[10] W Wu T R Lin and A C C Tan ldquoNormalization and sourceseparation of acoustic emission signals for condition moni-toring and fault detection of multi-cylinder diesel enginesrdquoMechanical Systems and Signal Processing vol 64 article 3837pp 479ndash497 2015
[11] D Yu M Wang and X Cheng ldquoA method for the compoundfault diagnosis of gearboxes based on morphological compo-nent analysisrdquoMeasurement vol 91 pp 519ndash531 2016
[12] A W Lees ldquoMisalignment in rigidly coupled rotorsrdquo Journal ofSound and Vibration vol 305 no 1-2 pp 261ndash271 2007
[13] Y D Chen R Du and L S Qu ldquoFault features of large rotatingmachinery and diagnosis using sensor fusionrdquo Journal of Soundand Vibration vol 188 no 2 pp 227ndash242 1995
[14] Y Yang and S Nagarajaiah ldquoBlind identification of damage intime-varying systems using independent component analysiswith wavelet transformrdquoMechanical Systems and Signal Process-ing vol 47 no 1-2 pp 3ndash20 2014
[15] K Yu K Yang and Y Bai ldquoEstimation of modal parametersusing the sparse component analysis based underdeterminedblind source separationrdquoMechanical Systems and Signal Process-ing vol 45 no 2 pp 302ndash316 2014
[16] Z Li X Yan XWang and Z Peng ldquoDetection of gear cracks ina complex gearbox of wind turbines using supervised boundedcomponent analysis of vibration signals collected from multi-channel sensorsrdquo Journal of Sound and Vibration vol 371 pp406ndash433 2016
[17] A Hyvarinen and E Oja ldquoA fast fixed-point algorithm forindependent component analysisrdquo Neural Computation vol 9no 7 pp 1483ndash1492 1997
[18] S I McNeill and D C Zimmerman ldquoRelating independentcomponents to free-vibration modal responsesrdquo Shock andVibration vol 17 no 2 pp 161ndash170 2010
[19] M J Zuo J Lin and X Fan ldquoFeature separation using ICAfor a one-dimensional time series and its application in faultdetectionrdquo Journal of Sound and Vibration vol 287 no 3 pp614ndash624 2005
[20] A Ypma and P Pajunen ldquoRotating machine vibration anal-ysis with second-order independent component analysisrdquo inProceedings of the 1st International Workshop on IndependentComponent Analysis and Signal Separation vol 99 pp 37ndash421999
[21] M J Roan J G Erling and L H Sibul ldquoA new non-linear adaptive blind source separation approach to gear toothfailure detection and analysisrdquo Mechanical Systems and SignalProcessing vol 16 no 5 pp 719ndash740 2002
10 Shock and Vibration
[22] Z Li X Yan Z Tian C Yuan Z Peng and L Li ldquoBlind vibra-tion component separation and nonlinear feature extractionapplied to the nonstationary vibration signals for the gearboxmulti-fault diagnosisrdquo Measurement Journal of the Interna-tional Measurement Confederation vol 46 no 1 pp 259ndash2712013
[23] L De Lathauwer J Castaing and J-F Cardoso ldquoFourth-ordercumulant-based blind identification of underdetermined mix-turesrdquo IEEE Transactions on Signal Processing vol 55 no 6 part2 pp 2965ndash2973 2007
[24] G Gelle M Colas and C Serviere ldquoBlind source separation atool for rotating machine monitoring by vibrations analysisrdquoJournal of Sound and Vibration vol 248 no 5 pp 865ndash8852001
[25] D Agrez ldquoImproving phase estimation with leakage minimiza-tionrdquo IEEE Transactions on Instrumentation and Measurementvol 54 no 4 pp 1347ndash1353 2005
[26] D L Davies and DW Bouldin ldquoA cluster separation measurerdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 1 no 2 pp 224ndash227 1979
[22] Z Li X Yan Z Tian C Yuan Z Peng and L Li ldquoBlind vibra-tion component separation and nonlinear feature extractionapplied to the nonstationary vibration signals for the gearboxmulti-fault diagnosisrdquo Measurement Journal of the Interna-tional Measurement Confederation vol 46 no 1 pp 259ndash2712013
[23] L De Lathauwer J Castaing and J-F Cardoso ldquoFourth-ordercumulant-based blind identification of underdetermined mix-turesrdquo IEEE Transactions on Signal Processing vol 55 no 6 part2 pp 2965ndash2973 2007
[24] G Gelle M Colas and C Serviere ldquoBlind source separation atool for rotating machine monitoring by vibrations analysisrdquoJournal of Sound and Vibration vol 248 no 5 pp 865ndash8852001
[25] D Agrez ldquoImproving phase estimation with leakage minimiza-tionrdquo IEEE Transactions on Instrumentation and Measurementvol 54 no 4 pp 1347ndash1353 2005
[26] D L Davies and DW Bouldin ldquoA cluster separation measurerdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 1 no 2 pp 224ndash227 1979
![[Alex] Construction Machineries [Instructor]](https://static.documents.pub/doc/80x56/56d6bec91a28ab3016938efa/alex-construction-machineries-instructor.jpg)
