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A fast algorithm for one-unit ICA-R

Qiu-Hua Lin a,*, Yong-Rui Zheng a, Fu-Liang Yin a,Hualou Liang b, Vince D. Calhoun c,d,e

a School of Electronic and Information Engineering, Dalian University of Technology, Dalian 116023, Chinab School of Health Information Sciences, University of Texas at Houston, Houston, TX 77030, USA

c The MIND Institute, Albuquerque, NM 87131, USAd Department of Electrical and Computer Engineering, University of New Mexico, Albuquerque, NM 87131, USA

e Department of Psychiatry, Yale University, New Haven, CT 06520, USA

Received 17 August 2005; received in revised form 18 September 2006; accepted 23 September 2006

Abstract

Independent component analysis (ICA) aims to recover a set of unknown mutually independent source signals fromtheir observed mixtures without knowledge of the mixing coefficients. In some applications, it is preferable to extract onlyone desired source signal instead of all source signals, and this can be achieved by a one-unit ICA technique. ICA withreference (ICA-R) is a one-unit ICA algorithm capable of extracting an expected signal by using prior information. How-ever, a drawback of ICA-R is that it is computationally expensive. In this paper, a fast one-unit ICA-R algorithm isderived. The reduction of the computational complexity for the ICA-R algorithm is achieved through (1) pre-whiteningthe observed signals; and (2) normalizing the weight vector. Computer simulations were performed on synthesized signals,a speech signal, and electrocardiograms (ECG). Results of these analyses demonstrate the efficiency and accuracy of theproposed algorithm.� 2006 Elsevier Inc. All rights reserved.

Keywords: Independent component analysis; Constrained independent component analysis; One-unit independent component analysis;Reference signal

1. Introduction

Independent component analysis (ICA) is a signal processing technique that aims to recover a set ofunknown mutually independent source signals from their observed mixtures without knowledge of the mixingcoefficients [8]. It has been increasingly used in blind source separation [6,13,30], feature extraction [18,4,12],and signal detection [26,34]. In most cases, ICA recovers all the source signals simultaneously. However, forsome source separation problems such as noise cancellation, biomedical signal processing, and financial anal-ysis [31,33,5,1], it is often preferable to extract only one desired source signal from the mixed signals. For

0020-0255/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2006.09.011

* Corresponding author. Tel.: +86 411 82922299; fax: +86 411 84708918.E-mail address: qhlin@dlut.edu.cn (Q.-H. Lin).

Information Sciences 177 (2007) 1265–1275

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example, only the speech signal is of interest when using ICA to extract a speech signal from a noisy mixture.As such, several one-unit ICA techniques have been introduced [2,11,14,10,7,20,21,17,29,3,23]. These methodscan be divided into two categories: one-unit ICA methods not using prior information and one-unit ICAmethods using prior information. One-unit ICA methods that do not use prior information typically extractall independent components one by one using a deflation process [11,14,10,7]. The major drawback of suchtechniques is that the order of the output signals cannot be pre-defined but is instead determined by theadopted contrast function and the initial conditions [20]. One-unit ICA methods which utilize prior informa-tion, on the other hand, are capable of extracting a single desired source signal by incorporating the priorknowledge into the ICA process [21,17,29,2,3,23].

Independent component analysis with reference (ICA-R) is an efficient one-unit ICA method utilizing priorinformation [20,21]. It incorporates prior information about the desired source signal as a reference signal toguide the separation process. As a result, the ICA-R algorithm extracts only the desired source signal, which isthe closest one, in some sense, to the reference signal [20,21]. The ICA-R algorithm has been successfully usedfor speech analysis [19]. However, a drawback of ICA-R is that it is computationally expensive. To reduce thecomputational complexity of ICA-R, a fast algorithm for ICA-R is proposed in this paper. An efficient imple-mentation of ICA-R is achieved by pre-whitening the observed signals and by normalizing the weight vector.Computer simulations were performed upon synthesized signals, a speech signal, and electrocardiograms(ECG). Results of these analyses demonstrate the efficiency and accuracy of the proposed algorithm.

2. One-unit ICA-R algorithm

2.1. Traditional ICA

Suppose that there exist M independent source signals s(t) = [s1(t), s2(t), . . . , sM(t)]T and N observed mix-tures of the source signals x(t) = [x1(t),x2(t), . . . ,xN(t)]T (usually N P M). The noise-free linear model ofICA is as follows [8]:

xðtÞ ¼ AsðtÞ ð1Þwhere A is an N · M mixing matrix that contains the mixing coefficients. The goal of the classical ICA is tofind an M · N unmixing matrix W such that M output signals

yðtÞ ¼WxðtÞ ¼WAsðtÞ ¼ PDsðtÞ ð2Þwhere P 2 RM·M is a permutation matrix, and D 2 RM·M is a diagonal scaling matrix. Consequently, thesource signals are recovered up to scaling and permutation.

2.2. One-unit ICA-R algorithm

The block diagram of the one-unit ICA-R algorithm proposed by Lu and Rajapakse in [21] is shown inFig. 1, in which x1(t), x2(t), . . . ,xN(t) are N observed mixtures of M source signals, y(t) is an estimated outputof the one-unit ICA-R algorithm, r(t) is a reference signal that carries prior information of a desired sourcesignal s*(t) but not identical to s*(t). The closeness between the estimated output y(t) and the reference signalr(t) is measured by e(y, r), which is treated as an a priori constraint to the ICA learning to find only one weight

r(t)

y(t)

xN(t)

x2(t)

ClosenessMeasure

WLearning

x1(t)

( , )y rε

Fig. 1. Block diagram of the one-unit ICA-R algorithm.

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vector w* (i.e., one row of the unmixing matrix W), so that the output signal y(t) = w*Tx(t) is equal to thedesired source signal s*(t) [21]. For simplicity, the time index t is omitted in the equations below.

A flexible and reliable approximation of the negentropy J(y) introduced by Hyvarinen in [16] is defined asthe contrast function of the one-unit ICA-R algorithm [20,21]:

JðyÞ � q½EfGðyÞg � EfGðvÞg�2 ð3Þ

where q is a positive constant, v is a Gaussian variable having zero mean and unity variance, G(Æ) is a non-quadratic function.

Suppose that the contrast function J(y) may have M local or global optimum solutions wi, i = 1, . . . ,M, togive M estimations of the M source signals. The closeness measure e(y, r) is defined to achieve its minimumwhen w = w* [21], i.e.,

eðw�Tx; rÞ < eðwT1 x; rÞ 6 � � � 6 eðwT

M�1x; rÞ

Hence, a threshold n can be used to distinguish the desired source signal s* from other source signals such thatg(w) = e(y, r) � n 6 0 is satisfied only when y* = s* among all the source signals. By considering g(w) as a con-straint to the contrast function (3), the one-unit ICA-R algorithm can be formulated using the framework ofconstrained ICA (cICA) [21,22] as follows:

maximize JðyÞ � q½EfGðyÞg � EfGðvÞg�2

subject to gðwÞ 6 0; hðwÞ ¼ Efy2g � 1 ¼ 0ð4Þ

The equality constraint h(w) is included to ensure that the contrast function J(y) and the weight vector w arebounded. With a slack variable Z, the inequality constraint g(w) can be transformed into an equality con-straint g(w) + Z2 = 0. By explicitly manipulating the optimum Z*, an augmented Lagrangian functionL(w,l) for the problem in (4) is given by [21]:

Lðw; lÞ ¼ JðyÞ � 1

2c½max2flþ cgðwÞ; 0g � l2� � khðwÞ � 1

2c hðwÞk k2 ð5Þ

where l and k are Lagrange multipliers for the constrains g(w) and h(w), respectively. c is a scalar penaltyparameter. A Newton-like learning algorithm can be derived by finding the maximum of the augmentedLagrangian function in (5) as follows [21]:

wkþ1 ¼ wk � gR�1xx L0wk

=dðwkÞ ð6Þ

where k denotes the iteration index, g is the learning rate, Rxx is the covariance matrix of the observed signalsx, and

L0wk¼ �qEfxG0yðyÞg � 0:5lEfxg0yðwkÞg � kEfxyg

dðwkÞ ¼ �qEfG00y2ðyÞg � 0:5lEfg00y2ðwkÞg � k

where �q ¼ �q, whose positive or negative sign is coincident with E{G(y)} � E{G(v)}, G0yðyÞ and G00y2ðyÞ are thefirst and the second derivatives of G(y) with respect to y, and g0yðwkÞ and g00y2ðwkÞ are the first and the secondderivatives of g(wk) with respect to y. The Lagrange multipliers l and k are learned by the following gradient-ascent methods [21]:

lkþ1 ¼ maxf0; lk þ cgðwkÞg ð7Þkkþ1 ¼ kk þ chðwkÞ ð8Þ

3. Fast algorithm for one-unit ICA-R

The fast algorithm is based on the following two considerations. First, there are two constraints in (4), theinequality constraint g(w) and the equality constraint h(w). The inequality constraint g(w) is exploited to incor-porate prior information of the desired signal into the ICA learning, whereas the equality constraint h(w) is

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used to bound the contrast function J(y) and the weight vector w. Indeed, J(y) and w can be bounded by nor-malizing the weight vector w. As a result, the ICA-R algorithm is simplified because we can omit the equalityconstraint h(w) and its corresponding learning parameters (such as the Lagrange multiplier k).

Second, the w learning algorithm in (6) involves the inversion of the covariance matrix Rxx, which is timeconsuming. If the computation of R�1

xx is avoided, the ICA-R algorithm can be further speeded up. Specifically,this can be achieved by data whitening preprocessing on the centered observed signals x:

~x ¼ Vx ð9Þwhere V is the pre-whitening matrix. Consequently, the covariance matrix of ~x equals unity, i.e.,

R~x~x ¼ Ef~x~xTg ¼ I ð10ÞThe fast ICA-R algorithm can thus be reformulated within the framework of cICA as follows:

maximize JðyÞ � q½EfGðyÞg � EfGðvÞg�2

subject to gðwÞ 6 0ð11Þ

Correspondingly, the augmented Lagrangian function L(w,l) for the problem in (11) is

Lðw; lÞ ¼ JðyÞ � 1

2c½max2flþ cgðwÞ; 0g � l2� ð12Þ

To find the maximum of L(w,l) in (12), a Newton-like learning method is also adopted to update the weightvector w:

wkþ1 ¼ wk � gR�1xx L0wk

=dðwkÞ ð13Þ

where

L0wk¼ �qEfxG0yðyÞg � 0:5lEfxg0yðwkÞg

dðwÞ ¼ �qEfG00y2ðyÞg � 0:5lEfg00y2ðwkÞg

Substituting (10) in (13), we get the simplified w learning algorithm:

wkþ1 ¼ wk � gL0wk=dðwkÞ ð14Þ

Then, it is followed by the normalization of the weight vector w:

wkþ1 ¼ wkþ1= wkþ1k k ð15ÞThe Lagrange multiplier l is still learned by the gradient-ascent method in (7), but the Lagrange multiplier kfor constrain h(w) is no longer learned.

In summary, compared with the original ICA-R algorithm, the fast ICA-R algorithm is improved in twoaspects: (1) by normalizing the weight vector w, the equality constraint h(w) and its corresponding learningdisappear; (2) by pre-whitening the observed signals, the inversion of the covariance matrix Rxx is avoided.As a result, the computation load for the fast ICA-R algorithm is considerably reduced.

Specifically, the fast ICA-R algorithm can be described as follows:

(1) Center the observed signals x to make its mean zero;(2) Whiten the observed signals x to give ~x;(3) Choose a proper scalar penalty parameter c;(4) Take a random initial vector w0 of norm 1, or just let w0 = 0;(5) Choose an initial value for the Lagrange multiplier l;(6) Compute the constant �q ¼ EfGðyÞg � EfGðvÞg;(7) Update the Lagrange multiplier l by lkþ1 maxf0; lk þ cgðwkÞg;(8) Update the weight vector w by wþkþ1 wk � gL0WK

=dðwkÞ;(9) Normalize the weight vector w by wkþ1 wþkþ1= wþkþ1

�� ��;(10) Until wT

k wkþ1

�� �� approaches 1 (up to a small error), otherwise go back to the step (6).

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4. Computer simulations and performance analyses

To verify the validity of the fast one-unit ICA-R algorithm, extensive computer simulations were carriedout, in which synthesized signals, a speech signal, and ECG signals were simulated. In the algorithm,G(y) = exp(�y2/2) was used [15]; and the closeness measure e(y, r) was specifically defined as mean squareerror, i.e., e(y, r) = E{(y � r)2}, and g(w) = E{(y � r)2} � n. The threshold n was initialized with a small valueto avoid local optima, and was gradually increased to ensure that the algorithm converged at the global max-imum [21].

The performance of the fast one-unit ICA-R algorithm and the original algorithm to estimate the desiredsignal was measured by the individual performance index (IPI), which was defined as follows:

IPI ¼XM

j¼1

pj

�� ��maxk pkj j

!� 1; k ¼ 1; . . . ;M ð16Þ

where pj denotes the j element of the global vector P = wTA. Besides, the accuracy of the recovered signal y

compared to the desired source signal s* was measured by the signal to noise ratio (SNR) in dB:

SNR ðdBÞ ¼ 10 log10

r2

mse

� �ð17Þ

where r2 is the variance (power) of the desired source signal, mse denotes the mean square error between thedesired source signal and the recovered signal, i.e., mse is the noise power.

4.1. Experiments with synthesized signals

Fig. 2(a) shows five synthesized signals S1 � S5 with data length of 2500 samples. Specifically, S1 is a Gauss-ian noise signal, S2 is a sub-Gaussian signal with kurtosis value of �1.24, S3 is a super-Gaussian signal withkurtosis value of 23.7, and S4 and S5 are two periodic deterministic signals. The mixing matrix was randomlygenerated with uniform distribution values between �1 and 1. Fig. 2(b) shows one example of five mixed sig-nals X1 � X5 of the five source signals S1 � S5 with the following mixing matrix A:

A ¼

�0:4552 0:9758 �0:5267 0:5944 0:3573

�0:8292 0:5234 �0:6208 0:4499 0:9031

0:7878 0:7528 �0:3645 0:0369 �0:5550

�0:7779 �0:7616 �0:2496 0:1996 �0:1068

�0:1901 �0:6265 0:2638 �0:4046 0:8645

26666664

37777775

A proper reference signal is important for the ICA-R algorithm to extract the desired signal. In the exper-iments, the reference signals for the periodic deterministic signals S4 and S5 were formed by sinusoidal signalshaving the same periods as those of the desired source signals [21], as shown in Fig. 3(a) and (b). The referencesignals for the random signals S1, S2, and S3 were constructed with sign operation to roughly give the signs ofthe most data samples of the desired source signals [21], as shown in Fig. 3(c)–(e). By using each of the fivereference signals, the fast one-unit ICA-R algorithm was performed on the five linear mixtures X1 � X5,respectively. As expected, five signals were extracted, as shown in Fig. 3(f)–(j). It is easy to see that the fiveextracted signals are almost identical (waveform preserved) to the five desired signals in Fig. 2(a).

To compare the performance of the fast one-unit ICA-R algorithm with the original one, all the experi-ments were performed twice (once for the fast ICA-R algorithm, once for the original algorithm). But the dif-ference between the output signals of both algorithms in terms of waveforms is hard to identify. Toquantitatively compare the performance of the fast algorithm and the original one, the IPI and the SNRdefined in (16) and (17) were computed for the separation results of both algorithms in the same experiments.Table 1 shows the results. It can be seen that the SNR and the IPI indexes are almost the same for the fastalgorithm and the original one. That is to say, both of them are capable of extracting the desired source signalsfrom the mixed signals with the same high SNR and the same low IPI. However, the running time consumed

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pyby both algorithms is different, as shown in Table 2. It can be found that the running time of the fast ICA-Ralgorithm is roughly half of that of the original algorithm.

By randomly generating the mixing matrix A, the simulation experiments were readily performed usingMonte Carlo runs. But it is shown that the extracted signals and the performance indexes for different runsare almost the same. Therefore, by pre-whitening the observed signals and by normalizing the weight vector,the computation complexity of the fast ICA-R algorithm is considerably reduced while the performanceremains essentially the same as the original algorithm.

4.2. Experiments with a speech signal

As mentioned above, the ICA-R algorithm can be used to extract a speech signal from its interferencesounds by exploiting a proper reference signal [19]. This was demonstrated with a speech signal and three envi-ronmental intrusions with 28,591 samples (sampled at 16KHz), as shown in Fig. 4(a)–(d) [9]. Fig. 4(a) showsthe speech signal of a man’s voice ‘‘I’ll willingly marry Marilyn’’. Fig. 4(b)–(d) show three intrusions, whichare random noise, noise bursts, and cocktail party noise, respectively. The power density spectra correspond-ing to the speech signal and three intrusions are shown in Fig. 4(f)–(i). Examining Fig. 4, we can find that thespeech signal has pitch frequency and its harmonic frequencies whereas the noises usually do not have [19]. Byutilizing this prior information, periodic rectangular pulses with pitch frequency of the speech signal can beused as the reference signal for extracting the speech signal. Fig. 4(e) shows the reference signal with only8000 samples out of 28,591 samples for identification. Comparing the power density spectra of the speech sig-

Fig. 2. The source signals and the mixed signals used in experiments with synthesized signals. (a): Five source signals: Gaussian noisesignal (S1), sub-Gaussian signal (S2), super-Gaussian signal (S3), two periodic deterministic signals (S4 and S5). (b): One example of fivemixed signals X1 � X5 of the five source signals S1 � S5.

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nal and the reference signal in Fig. 4(f) and (j), it is easy to find their similarity in that the reference signal alsohas fundamental frequency and harmonic frequencies.

By using the mixing matrix A generated in the same way as above, the mixtures of the speech signal andthree intrusions were easily obtained. One example of four noisy mixtures is shown in Fig. 5. The reference

Fig. 3. The reference signals and the extracted signals by the fast one-unit ICA-R algorithm in experiments with synthesized signals. (a):Sinusoidal signal having the same period as that of S4. (b): Sinusoidal signal having the base frequency of S5. (c)–(e): Impulse series havingthe same signs as those of S1, S2, and S3, respectively. (f)–(j): The extracted signals by the fast one-unit ICA-R algorithm corresponding tothe reference signals (a)–(e), respectively.

Table 1Comparison of IPI and SNR (dB) performance between the fast ICA-R algorithm and the original algorithm in experiments withsynthesized signals

S1 S2 S3 S4 S5

Fast IPI 0.10 0.04 0.002 0.20 0.05Algorithm SNR (dB) 37.07 27.06 30.19 46.47 35.04Original IPI 0.10 0.05 0.002 0.20 0.06Algorithm SNR (dB) 37.07 27.06 30.19 46.48 35.04

Table 2Comparison of time (s) consumed by the fast ICA-R algorithm and the original algorithm in experiments with synthesized signals

S1 S2 S3 S4 S5

Fast algorithm 0.0423 0.0415 0.0419 0.0422 0.0448Original algorithm 0.0760 0.0761 0.0788 0.0763 0.0814

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signal in Fig. 4(e) was easily constructed after the pitch frequency of the speech signal was estimated [27,28].Then, the fast ICA-R algorithm was used to analyze the noisy mixtures in Fig. 5; the waveform in Fig. 4(e) wasused as the reference signal. Fig. 6 shows the extracted speech signal. Comparing the extracted speech signal bythe fast ICA-R algorithm in Fig. 6 with the original speech signal in Fig. 4(a), their difference is hard toidentify.

Fig. 4. Comparison of power density spectra of a speech signal, three intrusions, and periodic rectangular pulses. (a): Speech signal.(b): Random noise. (c): Noise bursts. (d): Cocktail party noise. (e): Periodic rectangular pulses with the pitch frequency of the speech signalin (a). (f)–(j): Power density spectra corresponding to (a)–(e), respectively.

Fig. 5. One example of four noisy mixtures of the speech signal and three intrusions in Fig. 4(a)–(d).

Fig. 6. Extracted speech signal by the fast ICA-R algorithm from noisy mixtures in Fig. 5 with the reference signal in Fig. 4(e).

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The waveforms extracted from both algorithms (the fast ICA-R algorithm and the original ICA-R algo-rithm) were similar. For performance comparison, the IPI, the SNR, and the running time were computedfor both algorithms, respectively. Table 3 shows the results. Obviously, the experiments with a speech signalalso demonstrate that the fast ICA-R algorithm can achieve almost the same performance as the original onebut with a half shorter running time.

4.3. Experiments with ECG signals

Signal processing has become an invaluable tool for the analysis of ECG data [24,32]. The non-invasiveextraction of the fetal ECG (FECG) from the maternal cutaneous potential recordings presents benefit of

Table 3Comparison of performance including IPI, SNR (dB), and running time (s) between the fast ICA-R algorithm and the original algorithmin experiments with a speech signal

IPI SNR (dB) Running time (s)

Fast algorithm 0.0779 24.54 0.110Original algorithm 0.0779 24.57 0.193

Fig. 7. The ECG recordings, the reference signals, and the extracted signals by the fast one-unit ICA-R algorithm. (a)–(h): The 8-channelof ECG recordings obtained from a pregnant woman with data length of 2500 samples and time of 10 s. (i): The reference signal forextracting FECG. (j): The extracted FECG. (k): The reference signal for extracting MECG. (l): The extracted MECG.

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offering more detailed information about the fetus’s heart, compared to the conventional diagnostic tech-niques merely based on the fetal cardiac rate estimation [35]. The fast ICA-R algorithm can achieve such agoal by extracting the FECG from the maternal cutaneous electrode recordings with a proper reference signal.

Fig. 7(a)–(h) show 8-channel ECG recordings obtained from eight electrodes located on a pregnantwoman’s skin [25], each of which includes 2500 samples with time of 10 s. These cutaneous potential record-ings could be mixtures of the FECG, the maternal ECG (MECG), and the thermal noise from the electronicequipment. The FECG can be distinguished in the first five ECG recordings (see Fig. 7(a)–(e)). By using priorinformation about the FECG frequency, which is approximately 2.2 Hz, the reference signal for extracting theFECG can be formed by using impulse series with the same frequency, as shown in Fig. 7(i). Fig. 7(j) showsthe extracted FECG by the fast one-unit ICA-R algorithm. Similarly, by using the reference signal shown inFig. 7(k), which is impulse series with the approximate MECG frequency of 1.3 Hz, the MECG can also beextracted, as shown in Fig. 7(l).

As mentioned above, all the experiments were repeated by using the original algorithm, and the waveformsof the extracted signals by both algorithms are just the same. Since the mixing matrix A and the pure FECGand MECG signals are not available, the IPI and the SNR performance cannot be computed as above. But therunning time consumed by both algorithms can also be compared. Table 4 shows the results. Obviously,the experiments with ECG signals further demonstrate that the fast one-unit ICA-R algorithm can achievethe same satisfied results as the original algorithm in real applications, but its running time reduces approx-imately to half compared with that of the original algorithm.

5. Conclusion

In this paper a fast one-unit ICA-R algorithm is proposed by simplifying the constraints in cICA and byavoiding the matrix inversion in the weight vector learning. Specifically, one equality constraint and its cor-responding learning parameters are omitted by normalizing the weight vector, and the inversion of matrixis avoided by pre-whitening the observed signals. As a result, the computational complexity for the ICA-Ralgorithm is greatly reduced. Extensive simulations have been performed on synthesized signals, a speech sig-nal, and ECG signals to demonstrate the efficiency of the fast ICA-R algorithm. Both computer simulationstogether with performance analyses show that the speed of the fast one-unit ICA-R algorithm is twice fasterthan that of the original ICA-R algorithm, while the separation performance of the fast algorithm remains thesame as the original one.

Acknowledgements

The authors would like to thank the editor-in-chief and the anonymous reviewers for valuable comments toimprove the paper. We also thank Drs. Michael E. Brandt, Steven L. Bressler for English corrections and edit-ing. This work was supported by the National Natural Science Foundation of China under Grant No.60402013, No. 60372082, No. 60172073, the Liaoning Province Natural Science Foundation of China underGrant No. 20062174, and by the National Institutes of Health, under grants 1 R01 EB 000840 and 1 R01 EB005846.

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Table 4Comparison of time (s) consumed by the fast ICA-R algorithm and the original algorithm in experiments with ECG signals

FECG MECG

Fast algorithm 0.078 0.063Original algorithm 0.138 0.116

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