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Journal of Vibration and Control

DOI: 10.1177/1077546306063248 2006; 12; 395 Journal of Vibration and Control

Peter W. Tse, J. Y. Zhang and X. J. Wang Identification

Blind Source Separation and Blind Equalization Algorithms for Mechanical Signal Separation and

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Blind Source Separation and Blind EqualizationAlgorithms for Mechanical Signal Separation andIdenti�cation

PETER W. TSEJ. Y. ZHANGX. J. WANGSmart Asset Management Laboratory (SAM), Department of Manufacturing Engineering andEngineering Management, City University of Hong Kong, Hong Kong, China([email protected])

Received 28 June 2005� accepted 29 November 2005

Abstract: Many advanced techniques have been developed for diagnosis of machine faults caused by vibra-tion. They are effective if the inspected vibration is well isolated from interference caused by vibrationsfrom adjacent components. However, the components of manufacturing machines are numerous, small, andpacked closely together. Thus the signal collected by a sensor is the aggregate of vibrations from all nearbycomponents. This, coupled with noise, makes it nearly impossible to detect the anomalous vibration gener-ated by a particular component, especially those generated by small defective components. Recently, newsignal processing methods, such as blind source separation (BSS) and blind equalization (BE), have beenproposed to separate or recover the aggregated vibrations so that each source of vibration can be correctlyidentified. In this paper, a comparison study is presented. Some widely used BSS and BE algorithms havebeen compared to evaluate their performance in the separation of mechanical vibrations. Both simulated sig-nals and real vibrations generated by industrial machines were used to verify the effectiveness of BSS andBE. Their deficiencies have also been identified and improvements are proposed in the paper, so that theycould be effectively applied in the fault diagnosis of complex manufacturing machines.

Key words: Blind Source Separation, Blind Equalization, Blind Deconvolution, Machine Fault Diagnosis

1. INTRODUCTION

Machine vibration is a very complicated mixed information medium containing, amongstother things, information about the condition of the machine. Mechanical signal processingis crucial for extracting useful features from acquired vibration to determine the health statusof a machine. Recently, tremendous efforts have been made in this area with the aim ofachieving better accuracy in machine fault diagnosis. Numerous vibration-based machinefault diagnostic methods have been developed, such as exact wavelet analysis (Tse et al.,2004), Hilbert-Huang transform (Peng et al., 2005), singular value decomposition (Yang etal., 2003), and neural networks (Lai et al., 2004). Their effectiveness in the extraction of fault

Journal of Vibration and Control, 12(4): 395–423, 2006 DOI: 10.1177/1077546306063248

��2006 SAGE Publications

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396 P. W. TSE ET AL.

related vibration signals are promising as long as the acquired signals are not seriously inter-fering each other. Wavelet analysis has the ability to detect anomalous signals by applyingdecomposition to the inspected raw signal (Tse et al., 2001). It can detect any singular pointembedded in the raw signal, which enables the extraction of the incipient impulsive signal.However, the effectiveness of Wavelet analysis will be weakened if a high level of interfer-ence exists. Adaptive noise cancellation (Tse et al., 2003) possesses the ability to cancelundesired signal or noise in real time. However, it can only separate two dominant sources –the desired signal and undesired signal – from the raw signal. To separate signals generatedby multiple sources, a more efficient signal processing method is definitely needed.

Recently, the methods known as blind source separation (BSS) and blind equalization(BE) have been applied to separate or recover sounds, and have achieved promising results.The first method, BSS, is intended to separate sources using sensor observations that containmixtures of independent signal sources (Hyvärinen, 1999A). Due to a number of attractiveapplications in computer topography, wireless and mobile phone communications, BSS hasreceived increasing attention. Some well-known BSS algorithms are based on second-orderstatistics method (Bell and Sejnowski, 1995), information theory related Infomax method(Bell and Sejnowski, 1995� Roan, 2002), fixed-point method (Hyvärinen, 1999B), JADE al-gorithm (Cardoso and Souloumiac, 1993� Serviere, 2004), natural gradient algorithm (Amariet al., 1996� Cichocki and Unbehauen, 1996), and high order statistics method (Nandi, 1999).The second method, BE, is derived from BSS but aims to recover mixed signals that are cap-tured by a sensor at different time delays. It recovers the desired signal without any priorknowledge of the sources and the propagation channels. The desired signal can be recov-ered even with only one sensor (Godard, 1980). A major advantage of BE is that it doesnot require a training stage, which is essential in conventional signal equalization methods.They have some similarity in theory. In BSS, the sources are corrupted by the superpositionof the other sources, while in BE the source is corrupted by time-delayed version of itself(Bell and Sejnowski, 1995). That is, the purpose of BSS is to separate sources from theirunknown instantaneous mixture, while the aim of BE is to recover signal from a time-shiftedor convolutive mixture (Ypma et al., 2002).

Both BSS and BE possess their own merit in source separation. In general, BSS requiresmany sensors and that the signal generated by each source should be statistically indepen-dent. BSS has remarkable results as long as the mixed signals are a linear combination ofthe source signals and their propagation paths have no time delay. That is, the signals gener-ated from each source at a particular time will propagate to the sensors and be collected bythe sensors at the same time. BE, on the other hand, requires only one sensor and aims torecover captured signal in the presence of a time delay. However, the signal to noise ratio ofthe collected signals must be sufficiently high.[AQ?]

In the light of promising results, BSS and BE methods are also expected to be used forthe purpose separating or recovering mechanical vibrations generated by various mechanicalcomponents of a machine during its operation. In manufacturing machines, these compo-nents are numerous, small, and packed closely together. When the machines are running,each component will generate its own vibration, and these may interfere with each other inthe extraction of the original vibration generated by each component. Hence, the vibrationcollected from a sensor is the aggregate of vibrations generated by all nearby components.This, coupled with noise, makes it nearly impossible to detect an anomalous vibration gen-



BLIND SOURCE SEPARATION AND BLIND EQUALIZATION ALGORITHMS 397

erated by a particular component, especially those generated by small defective components.To enhance the accuracy in fault diagnosis, the aggregate vibration must be separated or re-covered so that each source of vibration can be correctly identified. Hence, the performanceof BSS and BE has been tested in manufacturing machines.

In this paper, a comparison study is presented. Five widely used BSS algorithms havebeen compared to evaluate their performance and usefulness in separating aggregated ma-chine vibrations. The eigenvector based BE algorithm tested, named EVA, was also appliedto temporal signals with different time delays. Both simulated signals and real vibrationsgenerated by industrial machines were used to verify the effectiveness of BSS and BE. Theresults show the performance and characteristic difference between the two blind separationtactics in the application of machine fault diagnosis by recovering fault related signals fromthe mixed signals. The deficiencies of BSS and BE have also been identified and improve-ments are proposed in the paper, so that they can be used more effectively in fault diagnosisfor complex manufacturing machines. The paper is organized as follows. In the next section,the basic models of BSS and BE are presented. In section 3, the algorithms of BSS that aresuitable for separating mechanical signals are determined based on our suggested method ofperformance assessment. In section 4, the model of BE most suitable for recovering machinefault related impulsive signals is established. Section 5 describes the set up and results ofthe experiments using artificially mixed signals. Section 6 presents the set up of experimentsusing real machines, the signals collection method used, and the results of those experiments.Section 7 provides the insights obtained from the results, including the deficiencies of bothBSS and BE, and suggested improvements. Concluding remarks and recommendations onthe application of BSS and BE to machine fault diagnosis are presented in section 8.

2. BLIND MECHANICAL SIGNAL SEPARATION

AND ITS POTENTIAL

The process of blind mechanical signal separation can be generalized as follows. Records ofsensor signals collected from a multiple-input/multiple-output (MIMO) nonlinear dynamicalsystem are examined, with the objective of finding an inverse system, termed a reconstructionsystem, or an adaptive inverse system, so that we can use it to estimate the primary sourcesignals (Hyvärinen, 1999A� Bell and Sejnowski, 1995). This estimation is performed on thebasis of the output signals and the collected sensor signals, as well as some prior knowledgeof the mixing system. Preferably, the inverse system should be adaptive in such a way thatit has some tracking capability in non-stationary environments. Instead of estimating thesource signals directly, it is sometimes more convenient to identify an unknown mixing andfiltering dynamic system first, and then estimate the source signals implicitly by applying asuitable optimization procedure. Such an approach is often adequate when the inverse systemis difficult to identify or the number of observations is less than the number of sources.In situations where the signals generated by a number of sources are simultaneously andlinearly mixed, then the process of blind signal separation can be applied to recover theoriginal sources by separating the captured observations using a suitable adaptive filteringsystem. The methods used to separate and estimate the original sources without knowing thetransmission channel and the source are BSS, blind signal extraction (BSE), and independent




component analysis (ICA). If the captured signals involve time delay, then the suggestedseparation methods are BE and Blind Deconvolution (BD).

2.1. Blind Source Separation

In BSS algorithms, the mixing and filtering processes of the unknown input sources s j�k�� j �1� 2� � � � � n�may have different mathematical or physical models, depending on specific ap-plications. In the simplest case, m mixed signals xi�k��i � 1� 2� � � � � n� are expected to bethe linear combinations of n unknown mutually statistically independent, zero-mean sourcesignals s j�k� with noise contamination. This can be written as

xi�k� �n�

j�1

hi j s j�k�� i�k� (1)

or in matrix notation

x�k� � Hs�k�� v�k� (2)

where x�k� � [x1�k�� x2�k�� xm�k�]T is a vector of sensor signals, s�k� � [s1�k�� s2�k��

� � � � sn�k�]T is a vector of sources, v�k� � [�1�k�� 2�k�� m�k�]T is a vector of additive

noise, and H is an unknown full rank m � n mixing matrix. It is assumed that the sig-nals received by an array of sensors (e.g. accelerometers, microphones) are weighted sums(linear mixtures) of primary sources. These sources are typically time-varying, zero-mean,and mutually statistically independent, but the characteristics and quantity of the sourcesare totally unknown. In other words, only the sensor vector x�k� is available and it is nec-essary to design an adaptive algorithm that enables the estimation of sources and the iden-tification of the mixing matrix H. To solve the problem using BSS, an n � m, full ranklinear transformation (separating) matrix W must be found, so that the output signal vectory�k� � �y1�k�� y2�k�� yn�k�

�Tcan be defined as:

y�k� � Wx�k� (3)

The components in vector x�k� must be independent to the greatest possible extent. Theproblem thus involves the identification of the mixing matrix H and/or separating matrix Wwith good tracking abilities.

2.2. Blind Equalization

The beauty of BE is that it can reconstruct the source signal through an unknown systemby using only one sensor signal. The process of BE is briefly outlined in Figure 1. Thesequence of the received signal x�k� is the result of applying convolution to the originalsource signal with the impulse response system and the addition of disturbance or noise. Inorder to reconstruct the source signal from the sensor signal and to obtain an output signal, aninverse filter (blind equalizer) is needed so that a close replica of the original source signal




Figure 1. Basic configuration of BE.

can be obtained as. Unlike the ordinary equalizer, the blind equalizer performs withoutaccess to the original input or any known training sequence or desired signal as used in otherconventional adaptive filtering.

The received or observed signal x�k� is given by:

x�k� � h�k� � s�k�� k� (4)

The output of BE, y�k�, can be obtained by:

y�k� � e�k� � x�k� (5)

y�k� �L�

l�1

e�l�x�k � l� (6)

Now, the objective is to derive a set of equalizer coefficients e�k� for BE from the sensorsignal x�k� to determine an inverse of the impulse response of the BE process.

2.3. Differences between Blind Source Separation and Blind Equalization

Though BSS and BE belong to the same group of blind source separation methods, theyare significantly different in their capabilities and criteria of application. BSS is commonlyapplied to separate multiple sources with the signals mixed linearly. The number of sensorsmust not be less than the number of sources. When employing BSS, it is expected that eachsensor will be located as close as possible to a particular source. Hence, the captured signalfrom each sensor is closely related to a source and each source is highly independent (ie,they are highly uncorrelated with each other). One must also expect that each source willtransmit its signal to the sensors through one propagation path without delay. Moreover, themixed signal must be invariable in time for quality separation results. In machine-generatedvibrations, correlation of vibrations could be high as there are many similar components in amachine. Multiple transmission paths due to numerous mechanical links make the problemvery complicated. With BE, the model of signal mixing is not a prime concern, but the signalto noise ratio can have a major effect on its effectiveness. The major difference from BSS isthat BE uses only one sensor, making it more cost-effective than BSS, although the trade-offis that BE can only separate the dominant source from the mixed signals. The main advantage




of BE is that it can separate mixed signals that contain a time delay. This property makesBE useful for machine fault diagnosis, which always involves time delay in the capturedsignals.

3. COMMON TYPES OF BSS ALGORITHMS

In this section, we briefly describe the five common types of algorithm involved in this study.More details can be found in the references given. These algorithms are well known and areagreed to be prominent representatives of BSS algorithms. Although many new neural-network based algorithms have been proposed for BSS recently, they are derived from thesefundamental algorithms. Few quantitative measures (such as those given in section 3.3 of thispaper) have been proposed to evaluate and compare the performance of these BSS algorithmsin mechanical applications.

3.1. Preprocessing of Data

Some BSS algorithms require a preliminary sphering or whitening for the signal x�k�. Evenwith some which do not necessarily need sphering, better convergence can be achieved us-ing sphered data. Sphering means that the observed signal x�k� in equation (3) is linearlytransformed to a signal d�k�:

d�k� � Qx�k� (7)

such that the covariance matrix D of d�k� equals unity: E{DDT } = I. This transformation isalways possible. Indeed, it can be accomplished by classical Principle Component Analysis(PCA). Besides sphering, PCA may allow us to determine the number of independent com-ponents (if m � n, for m mixed signals generated from n sources) provided that the noiselevel is low.

3.2. Methodology of Five Common BSS Algorithms

3.2.1. Second-Order Statistics Method

The most popular methods of finding a linear transform are based on second-order statistics(as in equation (3)). This means they use only the information contained in the covariancematrix of the signal vector x�k� to get the representation. The second-order statistics (SOS)can be applied to the data which follows the characteristics of Gaussian distribution. If thevariable x�k� has a normal or Gaussian distribution, its distribution can be completely deter-mined using SOS. Another reason for the popularity of second-order statistical methods isthat they are computationally simple. They often require only classical matrix manipulation.However, the BSS methods based on SOS may be unable to recover the source waveforms ifthe temporal information is ignored.

Examples of SOS based BSS methods include SVD (Singular Value Decomposition),SOBI (Second Order Blind Identification) and AMUSE (Algorithm for Multiple Unknown




Signals Extraction). SVD uses the singular value decomposition for a single time delayedcovariance matrix (Comon, 1994). SOBI is based on the ‘joint diagonalization’ of an arbi-trary set of covariance matrices (Bell and Sejnowski, 1995). AMUSE uses the sub-optimaltime delay p = 1 (Tong et al., 1991). These methods use only the time delayed covariancematrix. Hence, they are relatively fast in processing but sensitive to noise.

Considering two correlation matrices Rx(0) and Rx(� ) of observation vector x�k� with� as the time delay and m = n, we may obtain the linear transformation W which satisfiesequations (8) and (9) by simultaneously diagonalizing two symmetric matrices as:

WRx�0�WT � Im (8)

WRx��WT � Dn (9)

It is obvious that this is a generalized eigenvalue problem. The linear transformation Wis the eigenvector of matrix R�1

x �0�Rx��. Therefore, after completing the above procedures,the linear transformation W can be known and the output vector y�k� can be obtained fromequation (3).

3.2.2. Joint Approximate Diagonalization of Eigenmatrices

The algorithm of Joint Approximate Diagonalization of Eigenmatrices (JADE) can be con-sidered as a natural extension or generalization of the SOBI algorithm. By means of the(joint) diagonalization of cumulant matrices, reasonably good statistical performance can beachieved by involving all the cumulants of orders 2 and 4 while a fast optimization is ob-tained by the device of joint diagonalization (Cardoso and Souloumiac, 1993). The strongestpoint of JADE is that it works off-the-shelf (i.e., requires no parameter tuning) (Cardosoand Souloumiac, 1993). The JADE algorithm starts with the estimation of the distributionsof the independent sources, which is generally required by BSS algorithms. Cardoso etal. noted the benefits of optimizing cumulant for approximations of data, leading them topresent a number of approximations to information theoretic algorithms that operate on thesecond and the fourth order of cross-cumulants. All the information theoretic measures canbe calculated by operations on cumulants, e.g. variance and kurtosis as the second and thefourth order cumulants respectively. The advantage of operating on cross-cumulants is thatthe algorithms do not require gradient descent, thus avoiding problems related to conver-gence. The disadvantage of JADE is that the estimation of a complete set of fourth ordercross-cumulants requires storage of O�n4� cumulant matrices(Cardoso, 1999).

As mentioned previously, JADE uses second and fourth order cumulants. The secondorder cumulant is used to ensure the post-processed data is ‘white’ (i.e., decorrelated). Thisproduces a whitening matrix and the whitened sources. A set of cumulant matrices is esti-mated from the whitened sources. Cardoso et al. demonstrate that the separating matrix canbe estimated as V� � W, where V is a rotation matrix that makes the cumulant matrices asdiagonal as possible according to the JADE contrast function (Cardoso, 1999).

The JADE contrast function is the sum of squared fourth order cross cumulants from theset defined in equation (10).




��y� ��

i jkl �i ikk

�Qi jkl�2 (10)

The contrast function is an effective measure of mutual information between the cross-cumulants.

3.2.3. Fixed-Point Algorithm

Hyvärinen et al. have introduced a family of algorithms that are grouped under the epithetof ‘fixed-point algorithms’. The members of this family are differentiated firstly by thealgorithmic approach and secondly by the contrast function used. The key to all the variationsis to find independent components by separately maximizing the negentropy of each mixtureof signals (Hyvärinen, 1999B� Hyvärinen and Oja, 2000). Two mainly approaches are usedin the fixed-point (FP) algorithm: The symmetric approach, which uses a modified rule forthe update of W that enables simultaneous separation of all independent components, and thedeflation approach, which updates the columns of W individually, finding the independentcomponents one at a time. Either of these approaches can be used in almost all non-quadraticcontrast functions to provide estimates of negentropy.

For the sphered data, the one-unit FP algorithm has the following form:

W�k� � E�xg�W�k � 1�Tx�� E�g��W�k � 1�Tx�W�k � 1� (11)

where the weight vector W is also normalized to a unit norm after every iteration, and thefunction g is the derivative of the contrast function as stated in equation (10).

The most interesting feature of the FP family of algorithms is that the deflation methodcan be used to find independent sources one at a time. Hyvärinen et al. noted that the orderingof the discovered sources was determined by the contrast function. If using kurtosis as thecontrast function, the super-Gaussian sources tend to be found first (Hyvärinen and Oja,2000). Given that speech signals are strongly super-Gaussian, this suggests the possibility ofusing the FP deflation algorithm with kurtosis as the contrast function to find the first speechsource signal.

3.2.4. Natural Gradient Algorithm

Natural Gradient (NG) algorithm is a popular and simple neural-network gradient algorithm(Amari et al., 1996). It was derived by applying the natural gradient concept to an earlieralgorithm introduced by Bell and Sejnowski (Bell and Sejnowski, 1995). Algorithms suchas SANG (Self Adaptive Natural Gradient algorithm with nonholonomic constraints), NG-FICA (Natural Gradient – Flexible ICA), NG-OnLine (On-line adaptive Natural Gradient)and Pearson-opt (Pearson system optimized) are all classed as NG-based algorithms. Theupdate rule for the separating matrix W is:

�W � �I� g�y�yT�

W (12)




The notation g(y) means that the nonlinearity g(y) is applied to all the components ofthe vector y � Wx. If necessary, it would be possible to apply a specific nonlinearity gi�yi�to each component yi separately. The algorithm uses the nonlinearity function tanh. Thelearning parameter is usually a small constant. The main advantage of the NG algorithmis that it is simple, purely neural-network based, and derived by minimizing the mutualinformation which is an information-theoretic measure for statistical independence.

3.2.5. Higher Order Statistical Algorithm

Higher order statistical (HOS) algorithms use the information in the distribution of receivedsignal x(k) that is not contained in the covariance matrix. That is, the distribution of x(k)must not be assumed to be Gaussian all the information of Gaussian variables (zero-mean)is not contained in the covariance matrix. In general, higher-order cumulants or momentsare used to form HOS based functions. Well known HOS based methods include ERICA(Equivariant Robust Independent Component Analysis) algorithm, SIMBEC (SimultaneousBlind Extraction using Cumulants) algorithm and FOBI (Fourth Order Blind Identificationwith Transformation matrix).

ERICA (which is asymptotically equivariant in the presence of Gaussian noise) wasdeveloped by Cruces, Castedo and Cichocki (Cruces et al., 2002). This algorithm separatesthe signals from m mixtures of n sources (with non-zero kurtosis) in the presence of Gaussiannoise. ERICA is a quasi-Newton iteration that will converge to a saddle point with locallyisotropic convergence, regardless of the sources’ distributions. Whitening the signal forbetter convergence is not necessary with this algorithm.

The SIMBEC algorithm was also implemented by Cruces et al.. It performs simultane-ous blind signal extraction of arbitrary group sources from a large number of observations.They proposed a gradient algorithm that could optimize the separation criteria when the ap-proximate (or hypothetical) densities of the desired signals were a priori known. They alsoextended the algorithm to other contrast functions that do not require explicit knowledge ofthe sources’ densities. The necessary and sufficient local stability conditions of the algorithmare exploited to obtain a fast convergence.

FOBI algorithm is probably one of the simplest HOS algorithms for BSS. The algorithmis an extension of the AMUSE algorithm, and it uses the fourth order cumulants (quadrico-variance) matrices. The FOBI algorithm is recommended for use if the sources’ signals arevery sparsely distributed and the number of observations is large.

3.3. Performance Evaluation Criterion

For comparing the performances of the aforementioned five types of BSS algorithms in sepa-rating mixed mechanical signals, we now propose a set of evaluation criteria. The criteria arederived from the work of Westner (Westner, 1999), who examined the computational com-plexity and separation quality of BSS algorithms for instantaneously mixed sources. Thetwo main measures used are elapsed time T in MATLAB for indicating the computationalcomplexity, and the difference between the mixing and separating matrices for determiningthe separation quality. If separation is perfect, then the separating matrix is the inverse ofthe mixing matrix and the error distance (E1) will be zero. As E1 increases, the separating




matrix is progressively farther away from the inverse of the mixing matrix, indicating thatthe separation quality is correspondingly poorer. In practice, if the value of this performanceindex is less than 5%, then the algorithm yields a good separation performance, and valuesbelow 8% represent acceptable performance. Any value greater than 10% will be classifiedas poor separation. The difference between the mixing and separating matrices is calculatedbased on Nandi (1999)’s proposed equation with minor modifications:

E1 � 1

2n�n � 1�

n�i�1

��n�

j�1

��pi j

��max j

��pi j

�� 1

��

n�j�1

��pi j

��max j

��p ji

�� 1

��(13)

Where P � �pi j� � WH, pi j is the �i� j�-element of the global system matrix P and max j pi j

represents the maximum value among the elements in the ith row vector of P, max j p ji doesthe maximum value among the elements in the ith column vector of P.

Another measure for determining the separation quality is the Signal-to-Noise ratio(SNR) of the separated outputs. A final SNR around 15dB is classified as an excellent sep-aration algorithm, 6dB is considered effective, and 3dB to reflect that the algorithm requiresperceptible improvement (Westner, 1999). The measure of SNR is defined as:

SNR � 10log10

�E[s�k�2]

E[n�k�2]

�(14)

where E(.) is the mathematical expectation of the arguments, s�k� is the desired signal, y�k�is the source signal estimated by the algorithm, and n�k� � y�k�� s�k� is the difference, orunwanted signal. Here, s�k� and y�k� are assumed to have the same energy content.

However, in many cases, negative SNR may result, which cannot be used to indicatethe accuracy of the algorithm used. Therefore, we propose another equation to correct thedeficiency of the SNR criterion. Here, we use the correlation coefficient to indicate thedegree of resemblance between the original and reconstructed source waveforms. This canbe done using the covariance computation method:

SCC�s�k�� y�k�� cov�s�k�� y�k�� cov(s�k�)cov(y�k�)

(15)

Here, the better the separation, the closer this SCC parameter approaches to 1, that is,y�k� � s�k� and hence SCC�s� y� � 1 (i.e., an algorithm giving an SCC value of 0.9999has performed better than one which producing a value of 0.999). On the other hand, ifthe recovered source waveforms do not match the sources at all, then they are statisticallyuncorrelated, giving SCC�s� y� � 0, while negative SCC represents opposite phase. Alter-natively, we may use another relative function, NSCC � �10 log10�1 � �SCC��, to indicatethe performance.

In summary, the time T obtained by MATLAB is a relative index and considered to beaffected primarily by the inherent complexity of the algorithms and variation of the compu-tational resources. E1 in equation (13) provides an approximately normalized error betweenthe separated signals and the original sources, while SNR and SCC represent the deviation insimilarity of the separated signals and the original sources. Especially, the SCC can provide




information on the phase change but does not consider the difference in signal power. Hence,the use of more than one performance criterion is suggested for a comprehensive comparisonif computation cost and time are not a prime concern.

4. BLIND EQUALIZATION

Blind equalization was first proposed by Sato for applications in digital communication(Sato, 1975). Its fundamental idea is to derive the characteristics of an equalizer from re-ceived signals without knowledge of the sources or andthe input channels. Three differentapproaches have been proposed for BE. The first approach obtains the characteristics bymaximizing the cost function that is defined in advance by the means of an adaptive algo-rithm (Sato, 1975). The second approach uses the high order spectrum to directly recover thecharacteristics of a source or a channel with high order cumulants (Cadzow, 1996). These cu-mulants can be calculated from the amplitudes and phases of the high order spectra generatedfrom the received raw signal. The key is to establish equations using the relationship betweenthe high order cumulants and the parameters of the channel. The desired characteristics canthen be obtained by resolving the equations. The last approach combines the capability ofneural networks with either the cost function or the high order cumulants (Tong, 1995). De-sired characteristics can be found by using neural networks to maximize or minimize the costfunction.

4.1. Establishing a Model for Fault-related Impulses

When high-frequency fault-related impulses occur in the vibration spectrum of an inspectedcomponent in a rotating machine, it can be assumed that the operating condition of the in-spected component is deteriorating. In such a case, the component is subjected to a seriesof periodic impulsive forces F�t�. The force’s function can be expressed with Fourier seriesas:

F�t� � F�t � T � � a0

2�

��n�1

�an cos n0t � bn sin n0t� (16)

where T is the period of impulses and 0 � 2��T is the basic frequency. The vibrationmodel of a linear system with a single degree of freedom is often expressed as:

Mx�� Cx

� � Kx � F�t� (17)

where M is the mass, C is the viscosity, K is the spring constant, and x is the displacement ofthe system. Using the principle of linear addition, the response of the system over time canbe rewritten as:

x�t� � a0

2k�

��n�1

an cos�n0t � n�� bn sin�n0t � n�

k�1� n2�2�2 � �2�n��2

(18)




where � � 0

n, n �

k

m, � � c

2mn, and n � arctg

2�n�

1� n2�2 .

Equation (18) is a typical forced vibration when the system is excited by periodic im-pulses. Incipient impulses usually occur in short intervals. A single impulse can be ap-proximated by a linear combination of several unit pulses. Assuming that the impulse is arectangular pulse P0, and the subjected system has no resistance, then the temporal responseof the system is

x�t� ��

P0�1� cosnt�k 0 � t � t1

P0[cosn�t � t1�� cosnt] t � t1

(19)

Based on equation (19), the vibration response to an impulsive fault exerted on the inspectedcomponent is an oscillation with a period T, a natural frequency n, and an amplitude whichis controlled by the exerted impulse and the system as defined in equation (16).

The vibration generated by a defective component is often overwhelmed by noise orinterference. Even though a sensor is intentionally mounted on a particular component of arotary machine in order to collect the vibration generated by that component, the collectedsignal is an aggregate of a number of vibrations. These undesirable vibrations are generatedby other components that are adjacent to the inspected component, and transmitted to thesensor via mechanical linkages or paths. Therefore, the collected signal is not from only onesource, but from multiple sources. The characteristics of the propagation path are unknown,but it can be assumed that the small periodic impulses acting internally on the machine are apropagation of vibration waves. Their temporal responses can be modeled as time invariant,as defined in equation (19). The composite channel can be described by the causal (possiblymixed-phase) response h�k�. According to common theories in digital communications, acomposite channel can also be expressed with a similar block diagram to that shown inFigure 1.

The sequence of the received signal x�k� is the result of applying convolution to theoriginal input signal s�k� with the impulse response system h�k� and the disturbance ornoise ��k� as:

x�k� � h�k� � s�k�� k� ��

i

h�i�s�k � i�� k� (20)

As mentioned previously, the equalizer e�k� works as an inverse filter of h�k�. Unlikean ordinary equalizer, the blind equalizer performs without knowledge of the original inputor any known training sequence or desired signal (as would be used in any common adaptivefilter). As defined in equation (5), the equalized sequence y�k� is the result of convolutionof the received sequence x�k� with the equalizer e�k�. The objective of creating such blockdiagram and the model is to find the Finite Impulse Response (FIR) of the equalizer or theinverse filter e�k� that will make y�k� as close as possible to the delayed original signals�k � k0� defined in the equation of Mean Square Error (MSE) as:

MSE�e� k0� � E��y�k�� s�k � k0��2� (21)

where k0 is the delay of the input signal.




Figure 2. Basic configuration of EVA.

4.2. Algorithm for Blind Equalization

Blind equalization using the generalized eigenvector algorithm (EVA) is adopted here (Jelon-nek et al., 1997). This algorithm uses an additional virtual equalizer f �k�, which is similarto e�k� as shown in Figure 1, as a reference system to estimate its output z�k�. Hence, anadditional block can be inserted from the node x�k�, using the virtual equalizer f �k� as a‘reference system’. The output of this block will be z�k� as defined in equation (22). Theprocess of EVA is outlined in Figure 2.

z�k� � x�k� � f �k� (22)

The EVA takes the two-dimensional fourth order cross-cumulant as a maximum ‘cross-kurtosis’ quality function. A closed-form expression for the generalized eigenvector problemcan be derived by optimizing the above problem:

Czx4 eEVA � �RxxeEVA (23)

where Czx4 is the cross-cumulant matrix and Rxx is the auto-correlation matrix. The co-

efficient vector eEVA � [eEVA�0�� eEVA�l�]T is obtained by choosing the eigenvector of

R�1xx Czx

4 , which is associated with the maximum magnitude eigenvalue �. This is called the‘EVA-(l) solution’ as commonly defined in the field of BE. In addition, Jelonnek et al. haveproven that the magnitude of the combined impulsive response is ��k� � h�k� � f �k�. If��k� has reached its maximum value, wm � max��k��, only once, then the EVA-(l) so-lution that uses an FIR-(l) equalizer is unique. In practice, an iterative adjustment to thecoefficients of the reference system is required to guarantee the unique condition. Here, as-suming the coefficients of the FIR filter are fixed as f �0�� f �l� (arbitrarily), L is thenumber of received data samples, and I is a threshold value for stopping the iteration, thenEVA can be comprised by the following steps as described in the literature (see Jelonneket al., 1997):




1. Set the initial coef�cients f �0��k� � ��k � �l�2�� of the virtual equalizer for the ‘refer-ence system’. Set the iteration counter i to 0.

2. From the received sequence x�0�� x�L � 1�, estimate the (l + 1)�(l + 1) autocorre-lation matrix as Rxx � E�xk x�k �.

3. Use the received signal x�k� and the reference �lter coef�cients f �i� to compute z�k� �x�k� � f �i��k�, and then estimate the (l + 1)�(l + 1) 4th order cross-cumulant matrixCzx

4 � Czx4

4. Substitute Rxx and Czx4 as Rxx and Czx

4 into the equation (23).

5. Calculate the most signi�cant eigenvector eEVA of R�1xxCzx

4 when the magnitude of theeigenvalue � is at maximum.

6. Load e�i�EVA�k� into the reference system so f �i�1��k� � e�i�EVA�k� and increase the iterationcounter to i = i + 1.

7. Repeat steps from (3) to (6) until i � I.

5. RESULTS USING ARTIFICIALLY MIXED DATA

In order to obtain an accurate and quantitative measure of the performance of the chosenalgorithms for the separation of mechanical signals, four experiments using artificial mix-ing data were conducted. The first three experiments related to the performance of BSSalgorithms. The first experiment consisted of sinusoidal signals and vibration impulses, thesecond used artificially mixed real sounds captured by a multi-channel recorder, and thethird made use of four vibration signals generated by four mechanical components and thenmixed artificially. The fourth experiment was used to test the BE-based EVA algorithm, andused mixed signals including a sinusoidal waveform mixed with noise and also an impulsivesignal distorted by noise. The main reason for using artificially mixed signals is that weknow the original sources and the degree of mixing, and can therefore verify the accuracyof the results. Note that in the figures presented in the following sections, only the units ofthe observed signals are identified, and the amplitude of separated or recovered signals aredetermined using BSS and BE.

5.1. Separation Results of BSS on Artif icially Mixed Periodic Waveforms and Impulses

The occurrence of high frequency vibration impulses is usually a symptom of defective me-chanical components. These impulses are usually overwhelmed by vibrations generated bystructural or large mechanical components. This experiment used two impulsive signals (s1and s2) mixed with two sinusoidal waveforms with different periods (s3 and s4) as shownin Figure 3. Table 1 shows the results generated by different criteria for the separation ofthe artificially mixed signals using the various BSS algorithms mentioned in section 3. AllSOS based methods, including AMUSE, SVD, and SOBI, can completely separate the sinewaveforms s3 and s4 (shown by SCC3 and SCC4 = 1 or –1), although E1 is high (�10% intwo cases). However, they cannot separate the impulsive signals, as shown by poor values




Figure 3. Two pulse & two sine-wave signals.

Table1. Results for experiment 5.1.

Algorithm T E1 SNR1 SNR2 SNR3 SNR4 SCC1 SCC2 SCC3 SCC4SOS: AMUSE 0.30 0.1053 3.9915 3.9915 183.01 183.01 0.8455 0.8455 1.0000 1.0000SOS: SVD 0.05 0.1057 –6.396 –6.396 –6.021 –6.021 –0.8572 –0.8572 –1.0000 –1.0000SOS: SOBI 0.06 0.0859 5.7821 –6.2983 –6.0153 –6.0256 0.8894 –0.8894 –1.0000 –1.0000

JADE: JADE-op 0.09 0.1042 –6.0206 90.556 5.2881 –4.5882 –1.0000 1.0000 0.8479 –0.8479JADE: JADE-td 0.11 0.0113 –6.0133 73.531 18.855 26.132 –0.9998 1.0000 0.9990 0.9991

FP: FP 0.75 0.0000 –6.0206 140.00 –6.021 –6.021 –1.0000 1.0000 –1.0000 –1.0000NG: NG-OL 31.4 0.1628 2.0875 –6.6438 –5.5580 –5.3566 0.7860 –0.7860 –0.9839 –0.9839

HOS: ERICA 0.47 0.0000 –6.021 –6.021 120.53 –6.021 –1.0000 –1.0000 1.0000 –1.0000HOS: SIMBEC 0.14 0.0564 –6.021 –6.021 9.340 –4.9376 –1.0000 –1.0000 0.9473 –0.9473HOS: FOBI 0.03 0.1084 30.542 45.116 5.6415 –4.4432 0.9996 1.0000 0.8576 –0.8575

T: execution Time (sec). E1: Error distance (dimensional). SNR: Signal-to-Noise Ratio (db). SCC: SignalCorrelation Coefficient (dimensional)

of SCC1 and SCC2. The HOS methods SIMBEC and FOBI have greater ability to separatethe two impulsive waveforms than the sine waveforms, while ERICA performs equally wellon all the signals. The methods FP, ERICA and JADE-td performed better in separating allsignals than any of the other methods. The natural gradient based NG-OL fails to adequatelyseparate all of the waveforms. The SOS based AMUSE method achieved a very high SNRfor s3 and s4.

5.2. Separation Results of BSS on Four Artif icially Mixed Sound Sources

In this experiment, the task was made closer to reality by feeding each algorithm four realsound waveforms that had been artificially mixed. These four sound sources (shown in Fig-ure 4) have different temporal structures but are not entirely independent. The results of thisexperiment are summarized in Table 2. All of the algorithms performed well in separating the




Figure 4. Four typical sound signals.

Table 2. Results for experiment 5.2.

Algorithm T E1 SNR1 SNR2 SNR3 SNR4 SCC1 SCC2 SCC3 SCC4AMUSE 0.00 0.0409 –6.8552 27.819 21.763 15.607 –0.9966 0.9993 0.9967 0.9868SVD 0.03 0.0429 21.723 –5.7044 –6.2323 –6.1431 0.9966 –0.9992 –0.9961 –0.9859SOBI 0.02 0.0176 –6.8643 –5.6710 –6.2589 –6.1420 –0.9990 –0.9998 –0.9999 –0.9976JADE-op 0.05 0.2012 12.594 –5.9839 –6.5569 –6.0717 0.9721 –0.9680 –0.8686 –0.8700JADE-td 1.20 0.0326 –6.8125 –5.7359 25.403 –6.1305 –0.9990 –0.9977 0.9986 –0.9968FP 0.80 0.0119 38.392 –5.6720 24.270 –6.1553 0.9999 –0.9997 0.9982 –0.9999NG-OL 4.27 0.0383 –6.8382 17.019 27.907 –6.1226 –0.9999 0.9900 0.9992 –0.9833ERICA 0.97 0.0233 36.342 –5.6926 –6.2476 –6.1677 0.9999 –0.9988 –0.9990 –0.9988SIMBEC 0.64 0.0172 35.943 –5.6940 –6.2925 –6.1547 0.9999 –0.9990 –0.9989 –0.9998FOBI 0.03 0.0418 16.430 –5.6305 –6.2391 15.896 0.9886 –0.9994 –0.9986 0.9873

Definitions and units of T, E1, SNR and SCC are the same as Table1.

sound waveforms except the JADE-op algorithm. The dependence of the four sound wave-forms signals may cause the poor results. Of the methods tested, the FP algorithm achievedthe best separation performance, and also has a relatively modest computation cost. Mostof the algorithms (except JADE-op) achieved an average SCC value � 0.99 and E1 � 5%,demonstrating the reason why BSS is so popular for sound separation.

5.3. Separation Results of BSS on Four Artif icially Mixed Real Vibration Sources

Four typical mechanical vibrations captured by an ICP accelerometer were artificially mixedin this experiment. The four vibrations (see Figure 5) were generated by a gear (s1), a motor(s2), a pump (s3), and a bearing (s4). They were independently acquired using the sameaccelerometer with the same sampling frequency and gain but at different times and from




Figure 5. Four real vibration signals.

Table 3. Results for experiment 5.3.Algorithm T E1 SNR1 SNR2 SNR3 SNR4 SCC1 SCC2 SCC3 SCC4AMUSE 0.02 0.0228 24.608 –5.8458 24.303 30.237 0.9985 –0.9987 0.9985 0.9995SVD 0.05 0.2020 –5.4588 –3.7024 –5.8955 –5.2186 –0.9961 –0.6980 –0.9976 –0.7.75SOBI 0.05 0.0221 26.517 –5.8254 –5.9436 –5.2109 0.9990 –0.9992 –0.9978 –0.9995JADE-op 0.08 0.2636 4.9219 –5.8773 8.6138 –5.3548 0.8232 –0.9241 0.9324 –0.8082JADE-td 0.13 0.0551 21.435 –5.8329 –5.7097 17.884 0.9965 –0.9953 –0.9943 0.9921FP 0.78 0.0699 18.059 15.660 21.117 15.671 0.9923 0.9865 0.9961 0.9877NG-OL 1.41 0.3160 17.697 7.5100 1.9755 –5.9512 0.9918 0.9084 0.6092 –0.7110ERICA 0.73 0.0996 13.887 –5.8304 –5.8273 9.2498 0.9794 –0.9909 –0.9910 0.9388SIMBEC 0.28 0.0930 13.069 –5.8767 –5.7997 11.539 0.9752 –0.9904 –0.9970 0.9662FOBI 0.03 0.3030 –5.3457 6.3954 5.9393 2.1463 –0.8863 0.8841 0.8959 0.8005

Definitions and units of T, E1, SNR and SCC are the same as Table1.

different machines. Hence, the four vibrations could be considered as independent and dom-inant sources. Table 3 shows the separation results. The SOS algorithms AMUSE and SOBIachieved the best separation performance with E1 � 3% and SCC � 0.99. The FP andJADE-td algorithms produced less good results, but still with E1 � 7%. The SVD, JADE-op, NG-OL and FOBI algorithms failed to separate these signals. This experiment shows thateven algorithms derived from the same theory can produce results of very different quality,as illustrated by the group of SOS based algorithms.

5.4. Recovery Results using BE (EVA algorithm) on Artif icially Mixed SineWaveforms and Impulses with Noise

As mentioned previously, using artificially mixed signals for testing can be beneficial asthe input sources and noise level are known. The output results can be compared to the




Figure 6. Recovery of the distorted sine wave.

known sources, allowing the performance of each algorithm can be evaluated precisely. Inthis experiment, the first source (figure 6, left-hand side) is a sinusoidal signal generated bys�k� � sin�16�k�2000� and then distorted with white noise with a SNR of 6.4 dB to producethe observed signal shown in the left-hand middle diagram of figure 6. The second sourceis an impulsive signal created by h�k� � [–0.4, 1, 0.9, 0.12, –0.01] and also distorted usingsimilar noise (Figure 7). Note that the impulses contained in the original signal have beenseverely corrupted by noise, making the impulses difficult identify. The bottom diagrams ofFigures 6 and 7 are the results of deconvolution after applying EVA to the observed signalsof sources 1 and 2. It is obvious that the corrupted original signals of sources 1 and 2 havebeen mostly recovered with low level of noise, showing that BE based EVA is effective inrecovering both periodic and impulsive signals. It is worthwhile to note here that the phaseinformation contained in the two original signals remains indeterminate.

6. APPLICATION PERFORMANCE ON REAL SIGNALS

THAT ARE NATURALLY MIXED

In order to verify the performance of both BSS and BE algorithms in separating mechanicalvibrations, real vibrations generated by various mechanical components were captured si-multaneously. Because the mixing matrix and source signals could not be given explicitly inexperiments using real signals, the criteria developed previously for rating the performanceof BSS cannot be used. Instead, the separated results are directly compared with the ob-




Figure 7. Recovery of the highly corrupted impulses.

served signals. The first two sets of experiments were performed to test the effectiveness ofBSS algorithms, whilst the other two sets were used to test the BE based EVA algorithm.

6.1. Separation Results using BSS on Two Naturally Mixed Vibration Signals

The first experiment used two accelerometers to acquire the naturally mixed vibrations gen-erated by two vibration sources. The two vibration sources and two accelerometers (sensors)were placed on the experimental platform as shown in Figure 8. The arrangement is sym-metrical, so that S1 is the same distance from I that S2 is from V, and S1 is the same distancefrom V that S2 is from I. In Figure 8, the bottom source is a typical vibration generated by arotating mechanical component and behaves like a sinusoidal periodic signal. The top sourceis a series of impacts created by hammering the platform periodically. Figure 9 shows thetemporal waveforms of the observed signals captured by the top and the bottom sensors (left-hand-side top and bottom diagrams respectively), displaying the two naturally mixed sourcesignals with different amplitudes due to the positions of the sensors. The observed signalscan be treated as the result of a linear and instantaneous mixing. Hence, BSS algorithms canbe applied. The right-hand-side top and bottom diagrams of Figure 9 reveal the separatedsignals using the FP algorithm (similar results were also obtained using the AMUSE algo-rithm). Clear separation of the impulses can be found in the right-hand-side top diagram,but the result displayed in the right-hand-side bottom diagram does not separate the periodicvibration as expected. This is probably because signals generated in real environment maynot comply strictly with the assumptions required for BSS. Furthermore, noise, reboundedvibrations, and material intrinsic vibration made the case more complicated and so more




Figure 8. Experimental setup for two naturally mixed signals.

Figure 9. Two real vibration signals and the separation result.

difficult for the BSS algorithms to separate. Note that the ability to separate impactive sig-nals is valuable for machine fault diagnosis as mentioned previously, and so BSS algorithmsare still applicable in the field of fault diagnosis.

6.2. Separation Results achieved with BSS on Naturally Mixed Machine Vibrations

The second experiment used a machinery fault simulator to generate the required vibrationsand then collected these using three sensors (channels) in three different positions as shownin Figure 10. Sensor 1 was installed at the driving end of a motor, sensor 2 on the middlebearing house after the coupler of the shaft and sensor 3 on the bearing house at the end ofthe drive shaft. The shaft was rotating at 15.4 Hz. The vibrations were captured at a samplingfrequency of 8.192 KHz, and the efficient analysis frequency range was up to 3 KHz. Theleft-hand-side diagrams of Figure 11 display the temporal waveforms of the acceleration(unit: mm/sec2) of the vibration signals captured by the three accelerometers in order from




Figure 10. Machinery fault simulator.

Figure 11. Three real observed vibration signals and their separated signals.

top to bottom. The left-hand-side diagrams of Figure 12 show the spectra of the observedsignals corresponding to the three temporal observed waveforms shown in the equivalentleft-hand-side diagrams of Figure 11.

We adopted the AMUSE algorithm to separate the three observed signals. Becausealmost all BSS algorithms are sensitive to noise, we preprocessed these signals using a




Figure 12. The spectrum of three real vibration signals and their separated signals.

Savitzky-Golay FIR smoothing filter. The temporal waveforms of the separation results aredisplayed in the right-hand-side diagrams of Figure 11. The temporal waveform in the topdiagram reveals the usual vibration behavior of a motor driven shaft. The temporal waveformin the bottom diagram shows a lot of impulsive signals, which commonly occur in bearing-generated vibration. These behaviors are repeated in the spectra of the right-hand-side topand bottom diagrams of Figure 12. Note that in the bottom diagram, the spectrum includesa lot of high frequency vibration, which is typical for a faulty bearing frequency spectrum.From the separated results displayed in the right-hand-side diagrams of Figures 11 and 12,it is obviously that the first separated signal represents a shaft alignment problem due tothe relatively high amplitude ratio of the second and third harmonics of the shaft rotationalfrequency. The third separated signal reveals a bearing problem by showing a lot of energyin the higher frequency range. The second separated signal indicates a substantial vibrationcomponent at 200Hz but remains inconclusive as to the cause of this dominant vibration.

This experiment shows the abilities of BSS in machine fault diagnosis, but the interpreta-tion of results is difficult and some separation results are inconclusive. Moreover, the resultsfrom BSS have created some uncertainties in this example of vibration based machine faultdiagnosis. First of all, the order and phase of separated signals are ambiguous and not in linewith the three original observed signals. Secondly, some BSS methods cannot be applied toreal mechanical signal separation because the basic assumptions of BSS cannot be strictlyadhered to. The third major problem is in the scales of the recovered signals. As shown inthe results displayed in the right-hand-side diagrams of Figures 11 and 12, the scales of theseparated signals are totally in comparable with the original scales of the observed signals.All of these phenomena can seriously affect the accuracy of machine fault diagnosis.




Figure 13. Recovered signal from impulse series A.

6.3. Recovery Results achieved with BE on Naturally Mixed Signals

In order to demonstrate the performance of the BE based EVA algorithm in recovering im-pulses embedded in a vibration signal, two experiments were conducted. The experimentalplatform was similar to the one shown in Figure 8, but this time only one sensor was installed,at an equal distance from the two sources. Again, the first source was a series of impulsesgenerated periodically by striking the platform with a hammer often used for impact tests,and the second source was a sinusoidal type of waveform, which was acted as interference tothe collected impulses. Vibration signals were collected by the sensor every time the hammerhit the platform (and simultaneously triggered the sensor). Two types of impulses, namedimpulse series A and B, were created by hammering the platform with two different periodsand generating sinusoidal waveforms with two different amplitudes as shown in the top dia-grams of Figures 13 and 14 respectively. The recovered signals for impulse series A and Bare shown in the bottom diagrams of Figures 13 and 14 respectively. Note that both impulseseries A and B were both recovered with small residuals, demonstrating that the BE basedEVA algorithm is capable of extracting impulses from vibration signals that are naturallymixed and captured in real-time.

6.4. Identif ication of Bearing Fault-related Impulsive Signal Using BE

To further demonstrate the effectiveness of the EVA algorithm in identifying impulsive sig-nal generated by a defective bearing, an experiment was conducted on an underground trainmotor. The vibration generated by a motor bearing is often overwhelmed by the larger vi-bration generated by the motor. Hence, it is very difficult to extract the bearing vibration for




Figure 14. Recovered signal from impulse series B.

bearing fault diagnosis. This time, the EVA algorithm was used to recover the weak bearingvibration from the overall motor vibration. The top diagram of Figure 15 shows the overallvibration collected from a rotating motor. It can be seen that some impulsive signals exist inthe observed signal but have been modulated by the motor’s vibration. The observed signalwas modulated as a result of a slight misalignment of the motor’s shaft. The signal recov-ered by applying the EVA algorithm to the observed signal is shown in the bottom diagramof Figure 15. Note that the recovered signal (final signal) clearly shows the existence ofquasi-periodic impulses without the modulation. The time interval of these quasi-periodicimpulses was investigated and found to match the bearing outer-race defect characteristicfrequency. Hence, an outer-race defect was suspected. An overhaul of the motor revealedthat the motor’s bearing did have an outer-race defect and that the impulses were caused bythe defect. Without the help from EVA, the fault-related impulses would have been veryhard to distinguish from the overall observed signal. Hence, the BE based EVA algorithm iseffective in detecting incipient impulses caused by the occurrence of faults in the machine. Itis also worthwhile to note here that the amplitude of the recovered impulses are not the sameas in the original observed signals� this change is clearly not a desirable feature in vibrationbased fault diagnosis.

7. DISCUSSION OF POSSIBLE IMPROVEMENTS

In the light of the results shown in the previous section, the selected common types of BSSand BE methods cannot fully separate or recover the aggregated vibrations generated by ro-tating mechanical components. A number of shortcomings have been identified, mainly in




Figure 15. Observed bearing signal and its outer-race defective impulses revealed by EVA.

the indeterminacy of scaling/amplitude, uncertainty in order, high sensitivity to noise, and(for a BSS-based algorithm) the requirement for many sensors. These shortcomings may seri-ously affect the accuracy of fault diagnosis. For instance, the enlarged scaling/amplitude in theseparated signals may mislead the machine operators about the deterioration in the conditionof the running machine. Substantial improvements must be made to the algorithms beforeapplying BSS and BE algorithms to separate aggregated vibrations generated by complexand compact manufacturing machines, in order to achieve robust machine fault diagnosis.

The problem of scaling/amplitude indeterminacy is quite obvious. Ideally, the scales oramplitudes of the separated signals should be equal or relatively proportional to the ampli-tudes of the original sources. However, as shown in Figure 11, the scales of the observedsignals range from –0.002 to 0.002 up to –0.05 to 0.05, whilst the scale range of the sepa-rated signal is within –0.2 to 0.2. Even if the effect of mixed sources is included, the totalenergy in all three observed signals is not conserved in the total energy of the three sepa-rated signals. Such indeterminacy does not seriously affect applications in sound or voiceseparation, as one can simply adjust the volume of the separated sound or voice to hearthe difference. However, when separating vibrations generated from machines, it becomesa major discrepancy. Based on conventional knowledge, the signals generated by a defec-tive running machine should be larger than those generated by a machine running normally,and will increase continuously in magnitude as the defect deteriorates. However, with theexistence of amplitude indeterminacy, one cannot determine from the results whether themachine is normal or defective. Hence, this problem must be resolved prior to successfulfault diagnosis. We suggest three methods that may be attempted to solve the problem. First,the amplitudes of the separated signals could be re-normalized according to the amplitudesof the observed signals, so long as the total energy is conserved. Second, the amplitudes ofthe separated signals could be artificially adjusted so that they are relevant to the amplitudesof observed signals, provided some information about the signals captured by the sensors




is known. For example, blind separation medial signals, such as electroencephalography(EEG), are adjusted to be relevant to the amplitudes of signals received by the sensors be-cause the amplitude of one of the source signals is known. The third method is based onthe projection of amplitudes of the observed signals. Here, we propose a combined methodthat uses power preservation and Pasval theory to solve the amplitude problem. The pro-posed method is capable of correcting the amplitudes by means of guaranteeing that the totalpower accumulated from the power captured by the sensors is conserved before and after theapplication of BSS, that is

Py1 � Py2 � � � � � Pyn � Px1 � Px2 � � � � � Pxn (24)

where Pyi�i � 1� 2� � � � � n� represents the power of each separated signal si , Pxi�i �1� 2� � � � � n� represents the power of each received source xi , and n is the number of sen-sors used. Equation (24) guarantees that the power of signals captured by sensors (observedsignals) can be preserved in the power of separated signals. By using the combined method,one can decrease the discrepancy caused by the indeterminacy of amplitude. Hence, thehealth condition of the inspected machine can now be determined.

The next shortcoming which must be resolved is the uncertainty in ranking the order ofseparated signals to correspond with the order of the observed signals as shown in Figure 11.Obviously, the second observed signal as shown in the left-hand-side middle diagram hasbeen shifted to be the third separated signal as shown in the right-hand-side bottom diagram.Figure 12 shows the frequency spectra of the signals, hence, one can easily identify themost likely rank of the order of separated signals corresponding to the order to the observedsignals. However, in practice, unless one knows the order of the unmixed sources, it isvery difficult to determine the correct order of the separated signals. In reality, because thesources are usually naturally mixed, the order problem could be so serious that the orderof the separated signals could be completely different from the order of observed signals.In vibration-based fault diagnosis, the order problem could cause serious problems, as themachine operator may incorrectly conclude that a working component is anomalous andthe defective component is working normally. Correction methods may include the use ofvector distance or similarity to recover the correct order of separated signals. We propose theuse of correlation and covariance coefficients to indicate the resemblance measure betweenthe observed signals and the result of the separated signals. Based on these values, thecorrect order of separated signals could be indicated through the comparison of calculatedcoefficients between observed signals and separated signals.

One of the major criteria in using BSS is that the number of sensors used to collect themixed sources should be larger than or at least equal to the number of known independentsources. For example, in the medical use of BSS in EEG analysis, more than 100 sensorsare required. If adopting this criterion in machine fault diagnosis, the number of sensors re-quired might be extremely large as the vibration collected is usually composed of a numberof vibration signals generated from many components. Hence, such a scheme may result inhigh costs and increase the complexity of diagnosis. We propose the use of partial BSS tominimize the number of sensors required. In practical machine fault diagnosis, it is not nec-essary to fully separate the aggregated vibration as in voice separation. The vital informationactually required is the separation of vibrations into those generated by normal operating con-




ditions, and those caused by faults. Moreover, some prior information of vibration sourcescan be obtained instead of having a ‘blind’ situation. Hence, one can use some knowledgeto help in the process of separating anomalous machine operating conditions. Such revisedBSS is called partial knowledge based BSS (PKBSS). The benefits include minimizing thenumber of sensors used, and hence reducing the operating cost, simplifying the diagnosticprocess, and obtaining faster decisions. There are two research methodologies proposed forPKBSS. The first method is deduction from normal operating conditions. First, adaptivenoise canceling (ANC) (Tse et al., 2005) is employed to minimize interference or noise fromthe signals collected in normal operating conditions, and the ‘clean’ signals stored as ref-erences. Then, for signals collected afterwards, BSS deflation technology will be used tosubtract the reference signal from the later collected signals. As the normal signal has beenremoved from the collected signal, the remaining signals will have fewer independent com-ponents and be simpler to analyze with the BSS algorithms. If the proposed scheme works,then fewer sensors than the source numbers could be employed for collecting source signals.Moreover, changes or trends in the signal can be observed more easily after subtracting thenormal references. The second method requires collecting typical fault signals for the par-ticular type of machine and making them into templates. Then new BSS separation criteriacan be determined based on these signal templates. The criteria must guarantee that the sig-nal templates are independent signals. Based on these templates, a BSS separation networkcan be established using adaptive learning algorithms, such as neural networks. Because thelearned BSS is tailor-made for the inspected machine, fewer sensors can be used for signalscollected afterwards.

Both BSS and BE are sensitive to noise, which can reduce the accuracy of signal sepa-ration. Because real signals always contain noise, a proper method must be implemented toreduce its effect. Besides using ANC to minimize noise (Tse et al., 2005), another alternativeis to treat noise as an independent source. However, the source of noise may be difficult toidentify, and so a more comprehensive approach must be used to make BSS and BE lesssensitive to noise. Here, methods such as Savitzky-Golay FIR smoothing filter or waveletde-noise technology could be used as a preprocessor to identify the temporal and frequencybands of noise. Then the unbiased quasi Newton algorithm (Cruces et al., 2002) can be addedto BSS and BE to make them less sensitive to these bands of noise.

For the BE algorithms that separate signals in iteration steps, such as the EVA algorithmwe have employed in this study, the number of iteration steps and the complexity of compu-tation could be reduced by the introduction of a reference system. The iterative procedureconsists of calculating the blind equalizer impulse response and the outputs of the filters.The reference system would generate an implicit sequence of training (reference) data forsubsequent use in the iterative process. After a certain number of iterations (based on thereceived data samples), both the reference and the equalizer systems will have a set of co-efficients very close to an optimized equalizer. This method would make the computationless intensive. Another suggestion is the use of higher order statistics, such as the fourthorder cumulant, like kurtosis, to obtain the optimal number of iteration and the filter parame-ters. By using such an optimization process, the computation requirements can be reduced,while increasing the robustness of fault diagnosis. Currently, the authors have committed toresearch on optimizing BE processes, such as EVA or the super exponential method. Theresults will be reported later.




8. CONCLUSIONS

In this paper, the results of an intensive study on the application of BSS and BE based sig-nal algorithms to machine fault diagnosis have been reported. We compared experimentallymany well-known BSS and BE algorithms using situations involving separating or recov-ering simulated signals and mechanical vibrations that were artificially or naturally mixed.Qualitative criteria were designed and used to evaluate the performance of some commontypes of BSS and BE algorithms for simulated signals and vibrations generated by machines.Through the analysis of the results, both advantages and shortcomings of these algorithmshave been identified. Aimed at applications in vibration based machine fault diagnosis, anumber of improvements for BSS and BE algorithms have been proposed to reduce theirintensiveness in computation, increase the ease of interpreting the results, and enhance theaccuracy possible in fault diagnosis.

In the light of experimental results, a number of observations have been made. Thefirst conclusion is that the assumption of statistical independence of each source as requiredby BSS does not have to be strictly fulfilled. The slight looseness of this requirement en-courages the use of BSS as a signal separation tool for more applications, particularly invibration based machine fault diagnosis. The second conclusion is that the SOS algorithmsare suitable for separating common vibration signals, while the FP and JADE algorithmsare suitable for recovering impulsive signals of the sort that often occur at high frequencieswith defective bearings. A rule of thumb is that one must investigate the nature of possiblevibrating components (if partially known) and the characteristics of observed signals beforethe selection of an efficient BSS algorithm, not work as ‘blind’ as the name BSS suggests.The third conclusion is that the BE algorithm is constrained by the choices of filter length,sample number and iteration number. A long filter length tends to eliminate some of theperiodic components in the original signal. A large sample number will increase both theresolution of the result and the computational requirement. This issue should be taken intoconsideration in the practical application of BE.

There is no doubt that both BSS and BE are promising methods for vibration based ma-chine fault diagnosis. However, the shortcomings that are discussed in this paper must besolved for fault diagnosis to be robust. The authors have proposed a number of methods toimprove these shortcomings, and research has also been conducted to verify the effective-ness of the improvements. The components of complex manufacturing machines in manyindustries are numerous, small and packed closely together. The suggested improvementsfor BSS and BE algorithms are expected to improve the accuracy of signal separation inthese machines, and so the efficiency and accuracy of fault diagnosis.

Acknowledgement. The work described in this paper was fully supported by a grant from the City University of HongKong, Hong Kong, China (project no. 7001616).

REFERENCES

Amari, S., Cichocki, A., and Yang, H. H., 1996, “A new learning algorithm for blind signal separation,” AdvancesIn Neural Information Processing Systems 8, 752–763.

Bell, A. J., and Sejnowski, T. J., 1995, “An information-maximisation approach to blind separation and blinddeconvolution,” Neural Computation 7, 1129–1159.




Cadzow, J. A., 1996, “Blind deconvolution via cumulant extreme,” IEEE Trans. on Signal Processing 13, 24–42.Cardoso, J. F., and Souloumiac, A., 1993, “Blind beamforming for non-gaussian signals,” IEE Proceedings Part F

140, 362–370.Cardoso, J. F., 1999, “High-order contrasts for independent component analysis,” Neural Computation 11, 157–192.Cichocki, A., and Unbehauen, R., 1996, “Robust neural networks with on-Line learning for blind identification and

blind separation of sources,” IEEE Transactions on Circuits and Systems 43, 894–906.Comon, P., 1994, “Independent component analysis – a new concept?” Signal Processing 36, 287–314.Cruces, S., Castedo, L., and Cichocki, A., 2002, “Robust blind source separation algorithms using cumulants,”

Neurocomputing 49, 87–118.Godard, D.N., 1980, “Self-recovering equalization and carrier tracking in two-dimensional data communications

systems,” IEEE Transactions on Communications 28, 1867–1875.Hyvärinen, A., 1999A, “Survey on independent component analysis,” Neural Computing Surveys 2, 94–128.Hyvärinen, A., 1999B, “A fast and robust fixed-point algorithms for independent component analysis,” IEEE Trans.

on Neural Networks 10, 626–634.Hyvärinen, A., and Oja, E., 2000, “Independent component analysis: algorithms and applications,” Neural Networks

13, 411–430.Jelonnek, B., Boss, D., and Kammeyer, K. D., 1997, “Generalized eigenvector algorithm for blind equalization,”

Signal Processing 61, 237–264.Lai, W. X., Tse P., Zhang, G. C., and Shi, T.L., 2004, “Classification of gear faults using cumulants and the radial

basis function network,” Journal of Mechanical Systems and Signal Processing 18, 381–389.Nandi, A. K., 1999, Blind Estimation Using Higher-Order Statistics, Kluwer Academic Publishers, Boston.Peng, Z., Tse, P., and Chu, F., 2005, “A Comparison study of improved Hilbert-Huang transform and Wavelet

transform: application to fault diagnosis for rolling bearing,” Journal of Mechanical Systems and SignalProcessing 19, 974–988

Roan, M. J., Erling J. G. and Sibul, L. H., 2002, “A new non-linear, adaptive, blind source separation approach togear tooth failure detection and analysis,” Mechanical System and Signal Processing 16, 719–940.

Sato, Y., 1975, “A method of self-recovering equalization for multilevel amplitude-modulation systems,” IEEETransactions on Communications 23, 679–682.

Serviere, C., and Fabry, P., 2004, “Blind source separation of noisy harmonic signals for rotating machine diagno-sis,” Journal of Sound and Vibration 272, 317–339.

Tong, L., Liu, R. W., Soon, V. C., and Huang, Y. T., 1991, “Indeterminacy and identifiability of blind identification,”IEEE Transactions on Circuits and Systems 38, 499–509.

Tong, L., 1995, “Blind deconvolution estimation,” IEEE Transactions on Communication 43, 2986–2995.Tse, P., Peng, Y. H. and Yam, R., 2001, “Wavelet analysis and envelope detection for rolling element bearing fault

diagnosis – their effectiveness and flexibility,” ASME: Journal of Vibration and Acoustics 123, 303–310.Tse, P., and Lai, W. X., 2003, “Minimizing interference in bearing condition monitoring and diagnosis by using

adaptive noise cancellation,” in Proceedings of the 16th International Congress and Exhibition on ConditionMonitoring and Diagnostic Engineering Management, Vaxjo, Sweden, August 2003, pp. 739–748.

Tse, P., Yang, W. X., and Tam, H. Y., 2004, “Machine fault diagnosis through an effective exact Wavelet analysis,”Journal of Sound and Vibration 277, 1005–1024.

Tse, P., Lee, K., and Cheung, M., 2005, “New methodologies in traction motor fault diagnosis – the use of Wavelettransforms and Active Noise Cancellation,” in Proceedings of the IEE International Conference on RailwayEngineering – Development into the 21st Century, Hong Kong, China.

Westner, A. G., 1999, Object Based Audio Capture: Separating Acoustically Mixed Sounds, Master’s Thesis, MIT,MA.

Yang, W. X., and Tse, P., 2003, “Development of an advanced noise reduction method for vibration analysis basedon singular value decomposition,” Journal of NDT&E International 36, 419–432.

Ypma, A., Leshem A., and Duin, R., 2002, “Blind separation of rotating machine sources: bilinear forms andconvolutive mixtures,” Neurocomputing, 49, 349–368.



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