Kurtosis-Based Constrained Independent Component ... - IEEE Xplore

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 7, JULY 2014

Kurtosis-Based Constrained Independent Component Analysis and Its Application on Source Contribution Quantitative Estimation Jie Zhang, Zhousuo Zhang, Wei Cheng, Xiang Li, Binqiang Chen, Zhibo Yang, and Zhengjia He

Abstract— Aiming at finding the major vibration and noise sources of vehicles, a quantitative estimation method for source contribution using the kurtosis-based constrained independent component analysis (cICA) algorithm is proposed. First, the similarity between the ICs and the reference signals with given characteristics is described by a concise and effective closeness measurement function. Meanwhile, how to choose the reference signals and the choice of some other closeness measurements is discussed. Then, a widely used contrast function, namely, kurtosis, is modified by the closeness measurement to obtain an enhanced contrast function. The fixed-point iteration and deflation approach are employed to train the separating matrix. Then, the enhanced contrast function is therefore maximized and the kurtosis-based cICA algorithm is obtained. After that, the source contribution is quantitatively calculated by the reduced energy of the mixed signals in each extraction: the reduction of the energy in mixed signals corresponds to the contribution of the extracted IC. The correspondence relationship between the ICs and source signals can be obtained by prior knowledge. Finally, the effectiveness of the proposed algorithm is verified by numerical simulation and experiments. The results show that the proposed method has high accuracy in separating sources and quantitatively calculating the source contribution. Index Terms— Constraint independent component analysis (cICA), kurtosis, reference signal, source contribution.

I. I NTRODUCTION

M

EASUREMENT technique plays a very important role in mechanical engineering and machinery manufacturing. The measurement instrumentation is the information acquisition part of the mechanical system, like the eye of the human. The information collected by the measurement instrumentation can help the mechanical systems work properly. The study on vibration or noise source is one of the most popular research aspects of the mechanical systems. To measure or monitor the vibration or noise sources, some researches have been done in measurement fields. Laser vibrometry-based 2-D

Manuscript received July 5, 2013; revised October 19, 2013; accepted October 20, 2013. Date of publication January 2, 2014; date of current version June 5, 2014. This work was supported in part by the National Natural Science Foundation of China under Grant 51275382, in part by NSAF under Grant 11176024, and in part by the China Post-Doctoral Science Foundation under Grant 2013M532032. The Associate Editor coordinating the review process was Dr. Kurt Barbe. (Corresponding author: Z. Zhang.) The authors are with the State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2013.2293236

selective intensity method is studied for source identification in the reverberant field [1]. By means of laser Doppler vibrometry and acoustic intensity techniques, approach to measure the transverse vibration and acoustic radiation of automotive belts is proposed [2]. There are also signal processing methods along with measurement to identify [3] or monitor [4] the vibration or noise sources. Measurement of vibration or noise source is indispensable part of the research of the mechanical system. However, in our research of searching for the major vibration and the radiated noise sources of the automobile, it is found that under some conditions, the desired information is not easy to get merely with measurement technique and instrumentation. Generally speaking, the vibration sources are unobtainable by direct measure when source signals are mixed each other and coupled with structural and environmental noises. Thus, we need some supplementary approaches in addition to the measurement techniques to obtain the desired information. To seek the vibration and noise source and measure the source contribution, several methods have been studied recently. Conditioned spectral analysis and virtual source analysis are researched to find the vibroacoustical sources of noise in a passenger vehicle compartment, but these coherence-based methods are not suitable in low frequency when sources generated by engine are highly correlated [5]. In addition, the accuracy of sound holography contour map is low, and the source contribution value cannot be attained as well. If the signal processing method is taken as a postprocessing approach in addition to the measurement, maybe we can get the desired results. Costa uses discrete Fourier transform technique along with the ultrasonic sensors to measure the wind speed [6]. The signal processing methods enjoy widely studies in mechanical engineering [7]–[10]. These methods have the advantage that they can give the mechanical operation information without modeling the mechanical systems [11]–[14]. These methods can be very simple, fast, and effective. Owing to it is difficult to exactly model the structure of the automobile, or to get the comprehensive information of the sources and the transmission paths, the independent component analysis (ICA) algorithm, which can retrieve the sources and mixing modes without the knowledge of the sources and the transmission paths, has been treated as a promising method for the source identification and contribution estimation.

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ZHANG et al.: KURTOSIS-BASED CONSTRAINED INDEPENDENT COMPONENT ANALYSIS

ICA has been extensively studied in the last decades, and is widely applied to biomedical signal processing, communication systems [15], image processing [16], and so on. Tong discussed the indeterminacy of the ICA, and he pointed out that the ICA is unsolvable or has multiple solutions [17], [18]. Comon gave the definition of ICA and introduced contrast function to ICA from blind deconvolution. Further, he indicated that ICA could be realized by the optimization of contrast function [19]. After those, a lot of ICA algorithms are brought up. By means of the differential geometry and Riemann structure of the parameter space, Amari proposed the natural gradient ICA algorithm [20]. Using fixed-point iteration to optimize kurtosis, Hyvarinen attained a robust ICA algorithms called FastICA [21]. Then, he obtained a more robust version of the FastICA by a new contrast function: negentropy [22]. Benefiting from its high computational efficiency and convenience, FastICA has been widely applied. In our research, we find that the outcomes of FastICA are random, because the ICA is a stochastic method. Moreover, the measuring progress is interfered by the environmental noises and this does affect the separating progress. This problem is also pointed out in [23] and [24]. Since the prior knowledge of source signals can be obtained from the measurements of vibration and noise sources during the design and production process of the vehicle, we can use this information to improve the ICA algorithm. There are available ICA algorithms which make use of the prior knowledge or source signals: some of them mix the sources with observed signals, then the source signals mainly determine the coefficients of the separating matrix and make the genuine separating matrix hardly be computed; the others repeat numerous calculations and filter the separation results according to the reference signals, which could not get a reasonable separating matrix. These methods are not suitable to our research, because we need the separating matrix to estimate the source contribution. When comparing with the aforementioned approaches, the constrained IC analysis (cICA), which introduces the prior knowledge into the ICA algorithm in the form of constraints, has a better performance and enjoys widely applications [25]–[27]. Lu first proposed the cICA [28], [29], and he made use of equality constraint to make sure the separating vector have unit norm, and simultaneously he utilized inequality constraint to guarantee that IC has the same characteristics with the prior knowledge. Huang studied the cICA and gave a better inequality constraint. In addition, he discussed the advantages and disadvantages of several closeness measurements [30]. Lin replaced the equality constraint with normalization step, and adopted prewhitening processing to avoid complex computations required for matrix inversion [25]. Because the kurtosis, which is the numerical statistics that reflects the characteristics of the mechanical vibration signal distribution, is simple both in theory and calculation, the negentropy is replaced by the kurtosis to obtain a simpler and more computationally efficient algorithm. In this paper, a kurtosis-based cICA algorithm is proposed [27] and a quantitative calculation method for source contribution is brought up. First, a closeness measurement to describe the similarity between the ICs and the reference signals with given characteristics is presented. Some instructions

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on how to construct the reference signals are given. Some other choices for effective closeness measurement are also presented. Then, the traditional contrast function, the kurtosis, is modified by the measurement to obtain a new contrast function. The contrast function can guarantee the ICs have the desired characteristics besides being independent from each other. After that the new contrast function is optimized by fixed-point iteration, which is well known for efficiency and robust. In addition, then the kurtosis-based cICA algorithm is obtained. The deflation approach and the Gram–Schmitz orthogonal method are used to extract the ICs from the mixture one by one. The energy contribution of IC equals to the reduction in energy of mixtures when the IC is extracted. Using the correspondence relationship between the ICs and source signals, the source contribution is measured. Finally, the proposed method is verified by the numerical simulation and two experiments. All the results show that the proposed method is effective to restore the vibration sources (or ICs) and mixing modes, and can quantitatively calculate the source contribution. The rest of the paper is organized as follows: the derivation of the kurtosis-based cICA algorithm is given in Section II. In Section III, the theory of source contribution quantitative calculation is presented. The proposed method is verified by the simulation in Section IV, and it is also applied to two experiment cases’ validation in Section V. Conclusions are drawn in Section VI. II. K URTOSIS -BASED cICA A LGORITHM A. ICA Model ICA is a statistical and computational technique for blindly separating statistically independent sources from their linear mixtures at the sensors, generally using the techniques directly or indirectly involving high-order statistics. The linear instantaneous ICA model could be described by the following formula: xi =

l

ai j s j + n i

j = 1, 2, . . ., m, j = 1, 2, . . . l

j =1

x = As + N

(In a matrix form)

(1)

where x = (x 1 , x 2 , . . . , x m )T denotes an m-dimensional discrete-time signals which can be observed, x i is the i th mixed signal measured by the i th sensor; s = (s1 , s2 , . . . , sn )T denotes an n-dimensional discrete-time source signals, whose components are assumed mutually independent, s j is the j th source signal; N represents m-dimensional noise, n i is the noise of the i th mixed signal; ai j denotes the mixing coefficient and is also the element of A; and A represents a constant m × n mixing matrix to be computed. The problem tackled with in this paper is assumed noise free, which means N equals 0. The target of ICA is to estimate a constant separating matrix W, which enables us to estimate source signals by formula y = (y1 , y2 , . . . , yn )T = Wx

(2)

where yi is the estimation of i th source signal, i.e., the IC.

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The identifiability of the model can be assured by imposing the following fundamental restrictions (in addition to the assumption of statistical independence) as follows. 1) All the source signals si , with at most one exception, must be non-Gaussian. 2) The number of observed linear mixed signals m must be at least as large as the number of source signals, i.e., m ≥ n. 3) The matrix A must be of full column rank. Under these assumptions, the mixed signals can be essentially transformed into source signals (or ICs), whose waveforms are preserved. The signs and scales of the components are not determined, i.e., each component is estimated only up to a multiplying scalar factor. In addition, any orderings of the components are also not determined [31].

B. Novel Contrast Function The problem of ICA is formulated into an optimization problem of a specific function by Comon. The specific function is called contrast function. The kurtosis or its absolute value is widely used as the measurement of non-Gaussianity in ICA. According to the central limit theorem, the distribution of a sum of independent random variables tends toward a Gaussian distribution. In other words, a sum of two independent random variables usually has a distribution that is closer to Gaussian distribution than any of the two original random variables. Then, maximizing the non-Gaussianity of the random variables can get the IC. Moreover, the kurtosis is simple in both computation and theory. Computationally, the kurtosis can be estimated simply by the fourth moment of the samples (if the variance is constant). Theoretical analysis is simplified for the following linearity property of kurtosis: If x 1 and x 2 are two independent random variables, then it holds [32] kurt (x 1 + x 2 ) = kurt (x 1 ) +kurt (x 2 ) kurt (αx 1 ) = α 4 kurt (x 1 ) where α is a constant. The widely used kurtosis contrast function is defined as 2 J (y) = kurt(y) = E{y 4 } − 3 E{y 2 } wT z,

M {y, r } = ρ H (y)H (r) N H (y (i ))− H (y) H (r (i ))− H (r ) i=1 =

N N 2 2 H (y (i ))− H (y) H (r (i ))− H (r ) i=1

i=1

(5) where ρ H (y)H (r) is the correlation coefficient between H (y) and H (r ). H (y) and H (r ) are the mathematical expectation of H (y) and H (r ), respectively. N is the number of samples. The choice of H (y) can be y, the envelope of y, Fourier transform of y, or the envelope of Fourier transform. Hence, it is with H (r ). To make sure the measurement has the same convergence direction with the contrast function, we use the square of M {y, r }. Then, the contrast function in (3) is enhanced to J (y) = E{y 4 } − 3(E{y 2 })2 +β{M{r, y}}2

(6)

where J (y) is the enhanced contrast function; β is a factor to adjust the impact of the introduced measurement to the contrast function. In addition to the measurement in (5), there are several other measurements to evaluate the similarity between the ICs and the reference signals. Huang gave the following four measurements [30]: M1 {y, r } M2 {y, r } M3 {y, r } M4 {y, r }

= −E {yr } = E{(y − r )2 } = 1/(E {yr })2 = − (E {yr })2 .

(7) (8) (9) (10)

In the cICA framework of this paper, (7)–(10) must be negative. Here, the IC y and reference signal r is of zero mean and unit norm. It is easy to find the measurement in (7) is similar to measurement in (5) when both y and r are normalized. But when used, the measurement in (5) is similar to the measurement in (10). The measurement in (7), (9), and (10) can be explained as the correlation between the IC and reference signal, while the measurement in (8) is the mean square error.

(3) C. Kurtosis-Based cICA Algorithm

wT

is one row of separating matrix W, where y equals to z is the vector of x after whitening, and E {·} denotes the mathematic expectation. Notice that all the variables in (3) are of zero mean. It can be further assumed that y has been normalized for simplification, then E y 2 = 1. To take advantage of reference signals, the measurement between reference signals and ICs is defined as M {y, r }

follows:

(4)

where r is the reference signal, which is the simulation signal with the desired characteristics, or the source signal. The concise and effective measurement function is defined as

After the enhanced contrast function is obtained, the optimization method is needed to find the w which maximize J (y). By means of fixed-point iteration, the following equation is obtained:

3 T − 3w + 2β E wT z · r {E {z · r }} . w=E z w z (11) After each iteration, w should be normalized: w ← w w2

(12)

where ·2 represents the 2-norm. Repeat (7) and (8) until the convergence occurs, then one row of W is obtained, namely


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The normalization in (12) is also needed after the iteration. Note that the IC y and reference signal r must be of zero mean and unit norm. The steps of algorithm using the measurement in (7), (8), or (9) are the same with the proposed algorithm with the measurement in (5), except that the updating is replaced by the updating in (14), (15), or (16), respectively. Then, it is found that the iteration in (14) cost the least amount of calculation. The formula in (15) is also more efficient in computation than that in (11). The iteration in (16) is complex in calculation, but it converges probably faster than the previous ones for it has a large step size. Incidentally, the large step size may lower the robustness of the algorithm. When comparing with existing ICA algorithms, the kurtosisbased cICA algorithm and the underlying contrast function have several desirable properties. By involving reference signals, the convergence speed is even faster than FastICA, whose convergence rate is cubic [22]. For deflation approach and reference signals, we only extract the desired ICs to avoid complex computation. From (11), it can be seen obviously that the advantages are introduced by

2β E wT z · r {E {z · r }} . The effect of measurement is adjusted by β, with the growth of β, the influence is reinforced, and vice versa. When β is set to zero, the kurtosis-based cICA algorithm degenerates to FastICA. Fig. 1.

Flowchart of the kurtosis-based cICA algorithm.

D. Choice of the Reference Signals one IC is extracted. The ICA is completed after all ICs are found. To extract more ICs, the orthogonalization has to be employed before repeating the one unit algorithm to guarantee the IC to be extracted later is different from the former extracted ones. A simple and effective way is deflationary orthogonalization using the Gram–Schmidt method. More precisely, if p − 1 vectors w1 , w2 , . . . , w p−1 have been extracted, before the algorithm is adopted, the following orthogonalization should be done to w p : ⎫ ⎧ p−1 ⎨ ⎬ wp ← wp − wp, w j w j (13) ⎭ ⎩ j =1

where ·, · is the inner product. Normalization of (12) is also required. The flowchart of the kurtosis-based cICA algorithm with deflationary orthogonalization is illustrated in Fig. 1. Now we get the kurtosis-based cICA algorithm with the measurement in (5). Likewise, using the measurement in formula (7), (8), or (9), the following fixed-point iteration in (14), (15), or (16) are obtained, respectively: 3

− 3w + β {E {z · r }} (14) w = E z wT z

3

w = E z wT z − 3w − 2β E wT z − r {E {z}} (15)

3 3 w = E z wT z − 3w + 2β {E {z · r }} E wT z · r . (16)

How to choose the reference signals acts as the first and most important problem in the application of cICA. The principle of the choice of the reference signals is to guarantee that the IC with the desired signal characteristics can be extracted from the mixed signals. Lu used a series of pulses with small widths and same period as the desired source signals in the simulation as the reference signals [29]. Although this is a simple simulation case, it can be concluded that the relative waveform of the reference signal is not important. Meanwhile, the occurrence time of each pulse of the reference signal must be corresponding to that of the desired source signal. The similar results are given in [26] and [33]. James pointed out that the wave form of the reference signal should be closely matched to that of the desired source signal, while Wang indicated that the temporal feature soft he interest needed be captured. Therefore, it is important for the correctness of the frequency and the phase of the reference signal. The amplitude is not so important for the normalization in the kurtosis-based cICA. In the perspective of the theory, the best reference signal is source signal itself. In our research, the key point is how to calculate the quantitative contributions of the sources, rather than finding the faulty signals [26] or the biomedical signals [25], [28]. The vibration source signals can be easily obtained by the sensors. The sensors must be placed as close as possible to the source to minimize interference from other sources. Maybe there is some interference in the source signals, but these cannot affect the separating results. Moreover, the relative shape of the reference signals is not very important, but the temporal features need to be maintained.

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TABLE I PARAMETER VALUES OF S OURCE S IGNALS IN (21)

In our research, the source signals themselves performs as the best reference signals. However, without loss of generality, we generate the reference signal as the other researches. The details and practical operation are presented in the numerical simulation and experiment cases. III. E STIMATION OF S OURCE C ONTRIBUTION If the main vibration and noise sources are located by the kurtosis-based cICA algorithm, some measures, such as reducing the vibration of the main sources, or interrupting the transmission path between the main sources and the shells can be taken to reduce the vibration and noise for vehicles. We give out the estimation method of source contribution based on the results of the kurtosis-based cICA algorithm. The separating matrix W derived from the kurtosis-based cICA algorithm ensures wi (row of W) as an orthonormal basis of the R m . R m denotes the M-dimensional space. The properties of wi (orW) are given as follows: 1 if i = j T wi w j = δ i j = i, j = 1, 2, . . . , m (17) 0 if i = j A = W−1 = WT

(18)

where A is the mixing matrix, W−1 is the inverse of W, and WT is the transpose of W. Using the good properties of W, the following equation is obtained: x−i = x − x, wi wi

i = 1, 2, . . . , m

(19)

where x−i is the vector of x which subtract the contribution of yi (equals to wiT x), x, wi wi is the project of x on wi . The correspondence relationship between IC yi and source signal can be got from prior knowledge. Thus, source contribution can be quantitatively calculated by comparing the energy of x−i and x Ri = 1 −

x−i 22 x22

(20)

where Ri is the contribution ratio of source which corresponds to yi . Since wi (i = 1, 2, . . . , m) is the orthonormnal basis of Rm , the sum of all contribution ratios equals to 100%. IV. N UMERICAL S IMULATION AND ACCURACY A NALYSIS First, four artificial variables are employed to test the kurtosis-based cICA algorithm. It is the key points of the simulation that how to generate the reference signals and quantitatively calculate the source contribution. Then, the influence of the measurement accuracy to the proposed source contribution calculation method is discussed. A. Numerical Simulation Since most mechanical parts, such as the gears, bearings, pistons, are symmetric structures, the vibration signals from them contain a lot of periodic signals. When there are defects in the gears and bearings, the vibration signals are typical modulation signals or unilateral shock attenuation signals. The environmental noise is always white Gaussian. Thus, the sine

Fig. 2.

Waveforms of the source signals.

wave signal s1 , the modulation signal s2 , the shock attenuation signal s3 and the Gaussian noise signal s4 are used to simulate the independent source signals. The generating functions of the source signals are given ⎧ s1 = 2 cos (2π f1 t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨s2 = 2 cos (2π f 2 t) sin (2π f 3 t) 4 (21) ⎪ s3 = cos (2π f4 (t − 1.25i)) exp (−a (t − 1.25i)) ⎪ ⎪ ⎪ i=1 ⎪ ⎩ s4 = n (t). The parameters in (21) are listed in Table I. Then, the source signals are randomly mixed according to (22) x = A [s1 , s2 , s3 , s4 ]T

(22)

where A is the mixing matrix whose elements are draw from the standard uniform distribution on the open interval (0, 1), the superscript T denotes the transpose of a matrix. The waveforms of the first one second of the source signals are displayed in Fig. 2 and the waveforms of the first one second of the mixed signals x are shown in Fig. 3. The Fourier spectrums of the mixed signals are shown in Fig. 4. Before the proposed cICA algorithm is used to extract the source signals, we seek a way to get the reference signals. As mentioned in Section II, the source signals themselves are the best reference signals. However, to simulate the engineering application, here we give one method to construct the reference signals. Suppose we know the running state of the mechanical systems (obviously, this is known because the experiment is done at a given working condition), the operating frequency (30 Hz) is used to construct a square wave to extract the periodical signal. The phase of the square wave can be obtained by the feature extracted from the mixed signals. The construction of the reference signal for modulation signal


Fig. 3.

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Fig. 6.

Waveforms of the ICs by kurtosis-based cICA.

Fig. 7.

Waveforms of the ICs by SOBI-RO.

Waveforms of the mixed signals.

Fig. 4.

Fourier spectrums of the mixed signals.

Fig. 5.

Waveforms of the reference signals.

is similar. From the Fourier spectrums, it can be noted that there is a signal whose frequency is 24 Hz, so the sine wave with a frequency of 24 Hz is used to extract the source signal. No other obvious line spectrums are observed. So we need not to construct more reference signal. The waveforms of the first one second of the reference signals are shown in Fig. 5. In Fig. 5, r 1 is square wave with frequency of 30 Hz to extract the sine wave signal, r 2 is square wave with frequency of 54 Hz to extract the modulation signal, r 3 is square wave with frequency of 24 Hz to extract the shock attenuation signal.

The kurtosis-based cICA algorithm is used to analyze the mixed signals in Fig. 4. The closeness measurement in (5) is used and the step length is chosen as 1. The first one second of the separation outcomes of the kurtosis-based cICA algorithm are illustrated in Fig. 6. The first three ICs are extracted by the cICA algorithm with the reference signals, while the last one is obtained from the residual mixed signal. It is clear that the orderings of the ICs are the same with the reference signals and source signals. The same mixed signals are also analyzed by the famous Robust SOBI with Robust Orthogonalization (SOBI-RO) algorithm [34], [35]. The first one second of the results of the algorithm is shown in Fig. 7. Comparing Fig. 2 with Fig. 7, it is obvious the ordering of the ICs is random. The similar index SNRs is used to evaluate the performances of the two algorithms SNRs = 10 log10 (σ 2 /MSE)

(23)

where σ 2 denotes the variance of the source signals, and MSE denotes the mean square errors between the source signals and the ICs. When the IC is similar to the source signal, the MSE between them are small. So the algorithm with bigger SNRs gets a better result. The SNRs of both algorithms for each IC are computed and displayed in Fig. 8. From the figure, it is observed that the kurtosis-based cICA algorithm has better performance the SOBI-RO algorithm in the separation of the mixed signals in Fig. 3.

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Fig. 8.


SNRs of the two algorithm. TABLE II C ONTRIBUTIONS C ALCULATION D EVIATIONS (S IMULATION )

Fig. 9. Changes of the results according to the percentage of the noise in the observed signals. (a) Changes in correlation coefficients with the percentage of the noise. (b) Changes in measure deviations with the percentage of the noise.

Then, the source contribution is quantitatively calculated by the proposed method. First the outcomes of the kurtosisbased cICA algorithm are used to compute the source contributions. The residual mixed signals x−i are computed by (19), and then (20) is used to calculate the quantitative source contributions. The source contributions of the four sources are [32.18%, 24.42%, 7.89%, and 35.51%], respectively. The source contributions are also calculated by the proposed method based on the outcomes of the SOBI-RO method. The source contributions of the four sources are [34.01%, 22.68%, 10.20%, and 33.10%], respectively. To evaluate the performances of the algorithms, the real source contributions are also calculated. One of the simulation sources is set to zero, then the sources are still mixed by the same mixing matrix. The energy difference between the original mixed signals and the new mixed signals is the contribution of the source shut down. The real source contributions are [32.17%, 24.34%, 8.32%, and 35.91%], respectively. The deviations of both ICA algorithms are listed in Table II. From Table II, we can find that the proposed method has a high accuracy in source contribution. The source contribution method based on the cICA algorithm is much better with deviations 3% in the observed signals. Then, it is concluded that the source contribution method is reliable when the noise accounts for