Wireless Sensor Network, 2009, 1, 453-457
doi:10.4236/wsn.2009.15054 Published Online December 2009 (http://www.scirp.org/journal/wsn).
Copyright © 2009 SciRes. WSN
453
Research on Blind Source Separation for
Machine Vibrations
Weiguo HUANG, Shuyou WU, Fangrang KONG, Qiang WU
Department of Precision Machinery and Precision Instrumentation,
University of Science and Technology of China, Hefei, China
Email: wghuang@mail.ustc.edu.cn
Received August 3, 2009; revised August 18, 2009; accepted August 20, 2009
Abstract
Blind source separation is a signal processing method based on independent component analysis, its aim is to
separate the source signals from a set of observations (output of sensors) by assuming the source signals in-
dependently. This paper reviews the general concept of BSS firstly; especially the theory for convolutive
mixtures, the model of convolutive mixture and two deconvolution structures, then adopts a BSS algorithm
for convolutive mixtures based on residual cross-talking error threshold control criteria, the simulation test-
ing points out good performance for simulated mixtures.
Keywords: Blind Source Separation, Independent Component Analysis, Convolutive Mixtures, Machine
Vibration, Residual Cross-Talking Error
1. Introduction
Blind source separation (BSS) is a typical tool to recover
source signals from several observations usually pro-
vided by a set of sensors. Due to the lack of prior
knowledge of the source signals, generally, this method
considers an assumption of independence between the
sources. It has been successfully used in many fields,
such as biomedicine, telecommunications, speech proc-
essing, underwater acoustics [1–3]. But BSS methods
have seldom been used for monitoring or diagnosing the
mechanical devices, in many cases the signals obtained
by sensors consist of some useless signals, such as envi-
ronment noise, other mechanical devices [4–6]. In these
condition, using BSS as a preliminary step will reduce,
even remove the useless signals, which can significantly
improve the efficiency and accuracy of the condition
monitor and fault diagnosis.
At present, most of BSS works are related to the sepa-
ration of linear mixtures of sources. However, the vibra-
tion of mechanical devices is complex, and so is the
propagation medium. The mechanical vibration signals
are often convolutive mixtures [7].
Consequently, this paper mainly describes BSS for
convolutive mixtures and its application to mechanical
vibrations.
2. BSS
2.1. BSS Introduction
Blind source separation is a signal processing technique
by which unobserved signals, also called sources, are
recovered from the observation of several mixtures. The
term “Blind” includes two facts: both the source signals
and the mixing structure are unknown. In the present
research work, most are assuming the mutual independ-
ence of the sources. This is the fundamental basis of
BSS.
For condition monitoring and fault diagnosis, the ob-
served signals are usually the output of a set of sensors
and linear combinations of the sources. Just as Figure 1
shows.
Figure 1. Observed signals and sources.
W. G. HUANG ET AL.
454
In Figure 1, the S1, S2, S3,…Sq denote the sources and the
x1, x2, x3,…xp denote the observations (output of sensors).
When condition monitoring and fault diagnosing of me-
chanical devices, the P observations (output of P sensors)
are the linear combinations of the q sources. In this way,
the i-th observation (output of the i-th sensors) is:
1
() ()
q
iijji
j
x
astn t

(1) 1,2, 3,,ip
where aij is the linear combination coefficients, ni(t)
denotes the environment noise received by the i-th sen-
sor.
The noise may be considered as a source signal, in this
way, the mathematical model of BSS could be shown as
() ()
X
tASt (2)
where 12
()[ (),(),,()]
T
p
X
txtxt xt
12
()[ (),(),,()]
T
q
Sts tstst
denotes the p observed
signals, which are the available data. A is the unknown
p
×q mixing matrix, which denotes the unknown propa-
gation. denotes the q source sig-
nals, which include the noises. Here, generally,
p
q and
denotes the transpose operator.
[]T
Here, assuming the mixing matrix A is invertible and
the sources Si(t) (i=1,2,…q) are statistically independent.
The assumption of independence between the sources is
physically plausible because they have different origins
[8].
The kernel of BSS is to find a q
*p
separation matrix B
and the recovered signals are
()()() ()St BXt BAStCSt 
(3)
If matrix B could make the matrix C be an identity
matrix, it could be concluded that the source signals have
been separated perfectly.
The general model of BSS could be shown as Figure 2.
2.2. Model of Convolutive Mixture [9]
For condition monitoring and fault diagnosis, vibration
analysis involves a convolutive mixture because of the
propagation medium (structure of the system). The envi-
ronment noise may be considered as a source signal. The
general model of a convolutive mixture can be repre-
sented as in the Figure 3 for two source signals and two
observation signals (p=q=2) to be simplified.
Moreover, if the sensors are located near the source
signals, respectively, we could consider that the filters
Figure 2. BSS general model.
Figure 3. Two source signals and two observation signals
model.
A11 and A22 are equal to 1. In fact, it is significantly for
condition monitoring and fault diagnosis that the sensors
are as close as possible to the origins. In this way, we can
get a simplified model:
11 122
22112
()()* ()
()* ()()
x
nsnAsn
x
nAsnsn


(4)
Using the Z transform
11122
22112
()()() ()
()() ()()
XzSzA zSz
XzAzSzSz


(5)
and the matrix formulation
()()()
X
zAzSz
(6)
where
12
21
1
() 1
A
Az A
(7)
assuming that the filters of A(z) can be modeled by
Lth-order causal transverse filters, so that the matrix
are given by
()Az
1
0
() ()
L
k
ij ij
k
zakz
(8)
Formulation (5) may be expanded in the time domain
1
11 122
0
1
22112
0
()()() ()
()() ()()
L
k
L
k
x
nsn aksnk
x
naksnks
n


(9)
2.3. Separation Principle and Criteria
Theoretically, if the mixing matrix A is invertible, and
the separation matrix B is the inverse matrix of the mix-
ing matrix A, it could be reconstituted the source signals
perfectly. But, in fact, the mixing matrix A is unknown;
we could not obtain the separation matrix B by A di-
rectly.
In the case of the Figure 3, Herault [2] proposed a solution
Copyright © 2009 SciRes. WSN
W. G. HUANG ET AL.455
Figure 4. Recursive structure for the BSS.
Figure 5. Direct structure for the BSS.
based on a recursive architecture, which can be general-
ized in the case of convolutive mixtures modeled by FIR
filters as Figure 4: Recursive structure.
Also, it can be modeled by Figure 5: Direct structure.
In the Z-domain, the relationship between recovered
signals and observation signals is
~
11
12
~
21 2
12 21
2
()
1()
() 1
() ()
1
1()()
()
Xz
Bz
Sz
Bz Xz
BzBz
Sz









(10)
Therefore, the relationship between source signals and
recovered signals is
112 2112121
212121 122
12 21
2
1()() ()()()
() 1
()() 1()()()
1()()
()
BzAzAz BzSz
Sz
A
zBzBzAzSz
BzBz
Sz
 



 



(11)
For the Formulation (10),
If 1212
() ()
B
zAz and 21 21
() ()
B
zAz,
Then
()(), {1,2}
ii
Sz Szi
(12)
If 12
21
1
() ()
Bz Az
and 21
12
1
() ()
Bz
A
z
,
then
()() (),{1,2}
iij j
SzAzSzi j
(13)
In the case of the Formulation (13), the filters Bij(z) are
infinite impulse response, so only the case of the Formu-
lation (12) is valid. In practice, the mixing filters Aij(z)
are unknown, therefore, we must estimate them by a
method, which could be a stochastic iteration by maxi-
mizing the independence.
(1, )(, )[(()) (())],{1,2},[0,]
ijijn ij
bnkbnkuEfsngsn kijkL (14)
where bij (n,k) is the k-th coefficient of filter Bij at the
n-th iteration, un is a positive adaptation gain, ()
f
and
()
g
are non-linear functions, such as3
()
f
xx
and
()
g
xx
.
2.4. Performance Criteria
The separation performances two aspects:
Residual cross-talking error (RCTE), defined as
2
2
()
(,) 10lg()
ii
ii
i
Es s
RCTE ssEs









(15)
This could verify the quality of recovered signals, gen-
erally, when its value is less than –20db, we can consider
that the recovered signals are correct.
For the computing speed, this paper adopts the rule
based on RCTE threshold control criteria [10]
Figure 6. Source signals.
Figure 7. Observed signals.
Copyright © 2009 SciRes. WSN
W. G. HUANG ET AL.
456
Figure 8. Recovered signals.
Figure 9. RCTE between the recovered signals and the
source signals.
Figure 10. Frequency spectra of source signals, observed
signals and recovered signals.
12
RCTE RCTE
(16)
where α is the goal constant, its value is decided by the
separation efficiency and separation purpose.
3. Simulation and Results
As previously specified, the aim of BSS is to recover
unknown sources with the observations. The purpose of
this part is to illustrate the capability of BSS algorithms
to separate signals from rotating machine vibration. The
rotating machine vibration contains some character, in-
cluding transient impact, ambient noise [11]. From this
point of view, two simulation signals are generated as
below.
1
s=(sin(12 t)+0.4*sin(2 t))*(sin(100 t+0.5*sin(0.02 t)))

2
s()randn t
The mixture coefficients are acquired using a transfer
matrix defined in Equation (6), the mixing filter is
12
A=[0.8459 0.3561 -0.4625 0.0251 0.2413 -0.0866]
21
A=[-0.0599 0.0326 0.4426 -0.1977 -0.0748 0.7442]
The value of un is 0.0016.
The value of goal constant α is –42db.
In this way, we can obtain observed signals as Figure 7,
x1 and x2 are the mixture signals. The recovered signals
by the BSS algorithm are as Figure 8 shows. The algo-
rithm produces satisfactory separation results in Figure 9,
for recovered signals and , the RCTE can reach
18.58db and –23.59db respectively. It also shows satis-
factory results in Figure 10, the frequency spectrum of
recovered signals are almost the same as the sources. It
demonstrates that the transient can be extract from the
observed signals.
1
s
2
s
4. Conclusions
In this paper, we reviewed the basic theory of BSS for
convolutive mixtures, and then analyzed its application
to machine vibration, presented a BSS algorithm for
convolutive mixtures based on RCTE threshold control
criteria. The results of the simulation are favorable.
However, for real signals, improvements are necessary
for the mixture model and the algorithm. Presently, we
are trying to seek some other control criteria for BSS, do
research on its application to gearbox condition moni-
toring and fault diagnosis.
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Copyright © 2009 SciRes. WSN
W. G. HUANG ET AL.
Copyright © 2009 SciRes. WSN
457
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