Page 1

Journal of Computer Engineering

Research of Blind Signals Separation with Genetic Algorithm and Particle

Swarm Optimization Based on Mutual Information

Samira Mavaddaty

Department of Electrical and Computer Engineering

Babol Noshirvani University of Technology,

Babol, Iran

Email: s.mavaddaty@stu.nit.ac.ir

Abstract

Blind source separation technique separates mixed signals blindly without any

information on the mixing system. In this paper, we have used two evolutionary

algorithms, namely, genetic algorithm and particle swarm optimization for blind

source separation. In these techniques a novel fitness function that is based on the

mutual information and high order statistics is proposed. In order to evaluate and

compare the performance of these methods, we have focused on separation of noisy

and noiseless sources. Simulations results demonstrate that proposed method for

employing fitness function have rapid convergence, simplicity and a more favorable

signal to noise ratio for separation tasks based on particle swarm optimization and

continuous genetic algorithm than binary genetic algorithm. Also, particle swarm

optimization enjoys shorter computation time than the other two algorithms for

solving these optimization problems for multiple sources.

Keywords: Blind source separation, mutual information, high order statistics, Continuous

and Binary genetic algorithm, Particle swarm optimization.

1. Introduction

Blind source separation (BSS) has important applications in many area of signal

processing such as medical data processing, speech recognition and radar signal

communication [1-4]. In BSS, the source signals and the parameter of mixing model are

unknown. The unknown original source signals can be separated and estimated using

only the observed signals which are given through unobservable mixture [5]. In the

literature, the theory of BSS has been approached in several ways and as a result,

various algorithms have been proposed. For example independent component analysis

(ICA), principle component analysis (PCA), high order statistical cumulants and others

[6-8]. The most important and simplest of them is ICA as a statistical method that its

purpose is to find components of signal which have the most statistical independence.

ICA is based on random and natural gradient [9]. This algorithm is susceptible to the

local minima problem during the learning process and is limited in many practical

applications such as BSS that requires a global optimal solution. Also, the neural

networks have been proposed which their operation depends on an update formula and

activation function that are updated for maximizing the independence between

estimated signals [10]. These algorithms depend on the distribution of source signals.

Since this separation is executed blindly and there is no information about source

1 (2009) 63-76

77

Ataollah Ebrahimzadeh

Department of Electrical and Computer Engineering

Babol Noshirvani University of Technology,

Babol, Iran

Email: e_zadeh@nit.ac.ir

Page 2

Research of Blind Signals Separation…

signals, the distribution function of source signals should be estimated early.

Consequently, it leads to reduce the accuracy of problem solving. Thus, developing new

BSS algorithms on the basis of global optimization independent of gradient techniques

is an important issue [11-15]. The BSS problem is identified as a popular search among

researchers because it can work based on evolutionary algorithms such as continuous

genetic algorithm (CGA) and binary genetic algorithm (BGA), PSO and so on [16, 17].

It is obvious that GA and PSO are successful evolutionary algorithms that provide

heuristic solutions for combinatorial optimization problems.

In this paper, the BSS approach for linear mixed signals is studied to get the

coefficients of separating matrix by using PSO and both forms of GA. The operation of

these algorithms principally depends on the fitness function which in this paper uses

mutual information (MI) as a main criterion in information theory and high order

statistics (HOS) of kurtosis [18, 19, 20]. MI is a main quantity that measures the mutual

dependence of the two variables. Also the kurtosis is a simple and necessary criterion

for estimating dependency among signals [21]. This paper proposes the fusion of these

important criteria as a suitable fitness function for separation of different sources in

linear BSS model. Using this fitness function in evolutionary algorithms, it does not

need to have activation functions like what is required in neural network [22]. The

simulation results demonstrate the BSS scheme based on PSO and CGA is robust to

achieve global optimal solutions from any initial values of the separation system. These

results show high accuracy, suitable SNR and fast convergence of these evolutionary

algorithms than BGA. The BGA has bad convergence and lower values of accuracy and

SNR in source separation for more than three sources. The analyses show the

convergence of BSS using PSO is essentially faster than CGA and BGA for any number

of original signals. Thus, PSO is totally effective for this kind of optimization problems

especially in applications that needs high speed or low cost for time computations.

S.Mavaddaty & A.Ebrahimzadeh.

78

2. Linear Mixing Model and Separating Process

Assume that there exist n unknown signal

independent as possible. It is supposed that the source signals in linear model of BSS

are linearly mixed together With a matrix

A

is ,i 1,...,n

=

which are as mutually

n n

× that is unknown:

x As

=

(1)

Where

and n is the number of sources. The goal in solve of BSS problem is to discover the

source signals from x without knowing the nature of mixing matrixA. For doing this

task, separating matrixW should be found that it is

1

n

s [s ,...,s ]

=

and

1

n

x [x ,...,x ]

=

are n− dimensional source and mixed signals

1

WA

−

=

in ideal situation:

yWx

=

(2)

So that

model of BSS problem with sparse representation, which illustrated as Figure 1 includes

three procedures: an unknown mixing model, a recognition of mixing matrix and a

source signal retrieval process.

1n

y[y ,...,y ]

=

includes n− dimensional estimation of source signals. A general

Page 3

Journal of Computer Engineering

3. Preprocessing of BSS

1 (2009) 77-88

79

A. Centering

One of the most basic and necessary part of preprocessing is to center mixing signals

x so as to subtract its mean vector

{ }

m E x

=

that means convert x to a zero-mean signal

x [8]. This step should be executed because kurtosis basically obtains as follows:

{ }{ }

{ } { }

{ } { }{ }

(

{ }

E x

( )

()

()

{ }

()

{ }

2

()

)

()

2

2

42

2

2

22

4

2

2

E x3 E x

12 E x

E x

Kurt x

3 E x E x

4E x E x36 E x

−

=+

+

−

Figure 1. The general BSS flowchart consisting of unknown mixing model,

recognition of mixing model and source retrieval operation

The assumption of mixed data centering, makes easy the calculation of kurtosis. So,

the kurtosis can be computed by simple following formula:

{}

{}

E x

After estimating the mixing matrix W with centered data, the estimation by adding

the mean vector of x back to the centered estimates of s is completed. The mean vector

of s is given by

W m

, where m is the mean that was subtracted in the preprocessing.

()

4

2

2

E x

Kurt(x)

3

=− (4)

1

−

B. Whitening

Another useful preprocessing is to whiten the observed signals [23]. This means that

before considering the application of the ICA algorithm (and after centering), the

observed signal x is linearly transformed so that a new signal x obtains which is white,

i.e. its components are uncorrelated and their variances are equaled in unity. In other

words, the covariance matrix of x equals the identity matrix:

{}

E x.xI

=

T

(5)

The whitening transformation is always feasible. One popular method for whitening

is to use the eigenvalue decomposition (EVD) of the covariance matrix {

where E is the orthogonal matrix of eigenvectors of {

matrix of its eigenvalues,

1n

D diag(d ,...,d )

=

. Whitening can now be calculated by:

}

TT

E x.x EDE

=

}

T

E x.x

and D is the diagonal

Mixing

Model

1 s

2 s

3 s

A

Mixing

Model

Recognition

Source

signal

Retrieval

2

x

x

1

x

3

1

y

y

y

W

2

3

Page 4

Research of Blind Signals Separation…

x=ED E x=ED

S.Mavaddaty & A.Ebrahimzadeh.

80

-1 2T -1 2T

A

E As=As

(6)

The benefit of whitening is that it works based on the fact that the new mixing matrix

A is orthogonal. This characteristic of separating matrix reduces the number of

parameters needs to be estimated. Instead of estimating the

elements of the separating matrix, the new orthogonal separating matrix is estimated

that n(n 1) 2

−

contains degrees of freedom.

2

n parameters which are the

4. GA and PSO Algorithms for BSS

The algorithms that work based on evolutionary mechanism can be the best solution

for solving BSS problem through finding optimum and accurate coefficients of

separating matrix. According to these algorithms, Primary population can be converted

into a new population that independence among its components is maximized using a

suitable fitness function. Since GA and PSO intrinsically use evolutionary technique,

we take advantages of us as a successful and fast algorithm to jump out of the potential

local minimum.

A. Fitness function

There are two types of contrast function of BSS which are based on information

theory and high order statistics. The former methods include one of the mentioned

types. The fitness function proposed in this paper takes the fusion of two criteria,

kurtosis and mutual information. The kurtosis is a very simple and essential measure

that can be defined as:

•

Kurtosis < 3 for sub Gaussian signal

•

Kurtosis = 3 for Gaussian signal

•

Kurtosis > 3 for super Gaussian signal

According to central limit theorem that is totally practical in ICA, the distribution of

a sum of independent random variables tends toward a Gaussian distribution. Thus, a

sum of two independent random variables usually has a distribution that is closer to

Gaussian than any of the two original random variables. In BSS, if the kurtosis of

estimated signals is maximized and distanced from the kurtosis of Gaussian signal then

the reverse of the theorem is confirmed and independence among estimated signals is

guaranteed. So the fitness function can be defined based on the sum of the absolute

values of kurtosis in estimated signals. Another natural measure of dependence between

signals is inspired by information theory that is minimization of mutual information.

The mutual information I between n random variables

differential entropy is defined as follows:

()() ( )

12ni

i 1

=

That H is the entropy of mixed signals and

non-negative, and zero if and only if the variables are statistically independent that it

takes the form:

( )()()

ii

i 1

=

i y ,i1,...,n

=

using the concept of

n

I y ,y ,...,yH y H y

=−

∑

(7)

12n

Y[y ,y ,...,y ]

=

. The entropy is always

n

H Y P ylogP y

= −−

∑

(8)

Page 5

Journal of Computer Engineering

Thus, mutual information takes into account the whole dependence structure of the

variables, and not only the covariance, like PCA and related methods. So, the fitness

function can be defined as:

{ }

i 1

iii

J(y) [E y 3Ey H y H y ]

=

=−−+−

∑

Where

1n

y ,...,y are estimate of source speech signals. The dependence among the

estimated signals is minimized when Fitness is maximized. In this method, it is not

necessary to assume that the sources have the same sign of kurtosis, because the

absolute of fitness function is directly maximized. So the super Gaussian signals and

sub Gaussian signals can be separated from each other successfully.

1 (2009) 77-88

81

{ }

() ( )

422

n

(9)

B. Orthogonalization

The Orthogonalization plays a main and practical rule in BSS that the algorithm

would be completely defective without it. The estimate of coefficients using

maximization of fitness function to retrieve independent components is not enough.

Doing these steps until now, outputs of BSS algorithm are n similar speech signals that

are the estimate of source signal that its kurtosis is maximum. It should be mentioned

that algorithm does its task correctly because only when fitness function is maximized

that all n estimated signals have the analogy and maximum kurtosis. So the

orthogonalization is applied in order to avoid this problem. The orthogonal separating

matrix can be obtained by orthogonalization and satisfies (5). The orthogonalization is

applied to GA and PSO before fitting each population. When fitness function is

maximized, estimated signal is mutually independent as possible. Two main methods

for orthogonallization exist: Deflationary and Symmetric orthogonalization. Usually

Symmetric orthogonalization is used in ICA because of higher applicability and obtains

through the following formula:

T 1 2

W W.Real(inv(W.W ))

=

Doing Symmetric orthogonalization as the last necessary step for BSS, independence

among separated signals is guaranteed. The structure of the BSS based evolutionary

algorithm is shown in Figure 2.

−

(10)

5. Evaluating Criteria

In order to check the effectiveness of the proposed algorithm, the Euclidean distance

of the two vectors: the kurtosis of the estimated and source signals as error of proposed

method is investigated. The results of the separating process are better whatever this

criterion be less. Also, we utilize the SNR (signal-to-noise ratio) to confirm the

accuracy of Euclidean distance as evaluating criteria. We define SNR as:

2

i

i

2

SNR10log

Eytst

−

( )( )

()

ii

E (s (t))

=

(11)

Page 6

Research of Blind Signals Separation…

6. Simulation

S.Mavaddaty & A.Ebrahimzadeh.

82

In this experiment, nspeech signals are selected from TIMIT database and are

combined by an unknown mixing matrix with

2

n random values in uniform.

Figure 2. Structure of BSS based evolutionary algorithms

distribution in the range[ 1,1]

operator parameters, crossover and mutation probability per chromosome are

and

m

P 0.0025

=

, respectively. Learning factors in PSO are

simulation with binary genetic algorithm, each chromosome is encoded with eight bit

strings.

−

. The population size is N 80

=

. Regarding genetic

c

p

0.5

=

12

CC2

== . Also, in the

A. Separation of source signals

In this experiment, all of the three source signals are the speech signals that are super

Gaussian. The sample length is selected 14000. The mixing matrix A is randomly

chosen as:

0.08640.0524 0.0773

A 0.1585 0.99210.4158

0.65550.8930 0.3430

−−

−−

=−

Initialization

Produce initial population from n2 coefficients of separating matrix

Calculate estimated signal from Eq.2

Calculate fitness of each population from Eq.9

Produce new population and update it

Symmetric orthogonalization

Get optimum and best solution

Obtain separate signals

End

Meet stopping

criterion?

No

Yes

Page 7

Journal of Computer Engineering

Figure 3 represents 14000 samples from the source signals

kurtosises are 8.585, 4.7775, and 17.447, respectively. The mixed signals

shows in Figure 4. The separate signals

are shown in Figure 5-7.

Figure 3. Original source signalss

Figure 5. Separate signals yobtain based on CGA

Figure 7. Separate signals yobtain based on PSO algorithm

1 (2009) 77-88

83

123

x ,x ,x are

s ,s ,s that their

123

123

y ,y ,y using CGA, BGA and PSO algorithms

Figure 4. Signalsx mixed with unknown matrixA

Figure 6. Separate signals yobtain based on BGA

0 2000 40006000800010000 1200014000

-10

0

10

0 2000 400060008000100001200014000

-10

0

10

0 20004000 60008000100001200014000

-10

0

10

02000400060008000100001200014000

-10

0

10

02000400060008000100001200014000

-10

0

10

0200040006000 8000100001200014000

-10

0

10

0 2000 4000 60008000100001200014000

-10

0

10

0 20004000 60008000100001200014000

-10

0

10

0 20004000 60008000100001200014000

-10

0

10

0 20004000 60008000100001200014000

-10

0

10

0 2000 4000 60008000100001200014000

-10

0

10

02000 4000 60008000100001200014000

-10

0

10

0 20004000 60008000100001200014000

-10

0

10

0 2000 400060008000100001200014000

-10

0

10

0 20004000 600080001000012000 14000

-10

0

10

Page 8

Research of Blind Signals Separation…

The separated signals based on these three methods apparently are not different and

maybe seem that the solving the BSS problem based on these methods result in the

same accuracy. But, In addition to objective criterion, subjective criterion should be

concerned. Some parameters obtained by using these algorithms are shown in Table I.

According to these results it is obvious that CGA and PSO work better than that of

BGA with higher accuracy. Also, Table II gives the Euclidean distance criterion or error

in this paper and SNR of estimated signals. SNR values show that sources and obtained

signals have the greatest relationship and dependency. The results of other comparison

are shown in Table III that demonstrates the Correlation matrixes of separated signals

based these three algorithms. Also, this experiment redone on several speech signals

and mechanism of these algorithms are considered. According to this experiment, it is

clear that BSS based on CGA and PSO can separate up to three speech signals

successfully. Also, the applicability and efficiency of PSO for separating signals for

more than ten source signals considerably is better than CGA. Indeed PSO works better

than CGA in this condition with high speed and accuracy and fast convergence. These

characteristics are notable because many original sources usually exist in transfer

channel for data transmission that separation of them without losing data is main and

necessary. Figure 8 shows the error of applying CGA, BGA and PSO to speech signals

for different number of sources.

For more studies on the effectiveness of these algorithms in estimation of signals, the

speed of each algorithm with increasing of the number of original signals is considered.

The diagram of time computation versus the number of sources is shown in Figure 9.

According to this experiment, the speed of BSS using PSO noticeably is faster than

CGA and BGA for any number of original signals because PSO has no genetic

operators. It should be mention that time for converging of fitness function in BGA is a

few more than CGA because BGA has an addition step in its procedure. BGA firstly

converts each chromosome to an encoded binary string and works with the binary

strings to minimize the cost function and then decodes them to evaluate the fitness. But

CGA directly deals with chromosomes.

Figure 10 compares the best values of fitness function of the purposed algorithms for

blind source separation problem based on CGA, BGA and PSO. The abscissa

Table 1. Comparison among three algorithms

PSO

Parameter

20 Population

39.774 Optimum Fitness

39.773

Average Fitness

39.771

Worse Fitness

27 Iterative Time(sec)

17.448

Kurtosis(

1

y )

8.5853

Kurtosis(

2

y )

4.7774

Kurtosis(

3

y )

S.Mavaddaty & A.Ebrahimzadeh.

84

CGA

50

39.398

39.397

39.396

73

17.445

8.584

4.7763

BGA

50

39.288

39.169

39.093

90

17.443

8.584

4.7830

Page 9

Journal of Computer Engineering

Table 2. Comparison of Euclidean Distance criterion and SNR for three algorithms

1 (2009) 77-88

85

CGA

14.688

10.847

12.467

BGA

18.456

14.345

13.985

PSO

10.367

28.657

11.321

SNR of (

SNR of (

SNR of (

1

y )

y )

y )

2

3

0.0019951 0.002661 0.0019265 Euclidean Distance

Table 3. Correlation matrixes of separated signals for three algorithms

represents 200 iterations and y-axis represents the best cost of fitness function in

each generation. The turbulence of BSS based on BGA causes that this algorithm

converges with distortion. The BSS based on BGA in separation of multi source fails to

find the optimum with a population size of 80 in 200 generations. The BSS based on

CGA and PSO on the other hand finds the optimum without a doubt and usually finds

the optimum within less than hundred generations. As a result, CGA and PSO transform

each population to the better population using suitable operators based on proposed

fitness function correctly. It is definite the success of

Figure 8. The error diagram with increasing the

number of speech sources in simulation of CGA,

BGA and PSO

these algorithms depends on the definition of fitness function using kurtosis.

Figure 9. The diagram of time computations for

different number of source signals using CGA,BGA

and PSO in BSS problem

B. Separation of signals in the presence of noise

Industrial research is developing blind source separation algorithms to provide

enhanced separation of mixed signals or mixtures of signal plus noise or interference.

So, in this experiment an experimental demonstration by mixing signals with random

noise signal in interval [ 1,1]

−

is provided.

CGA BGA

PSO

1 0.00117 0.00059

0.00218 0.99999 0.00357

0.00136 0.00176 0.99984

0.99997 0.00286 0.00438

0.01255 0.99868 0.00202

0.00360 0.012029 0.99992

1 0.00040 0.00004

0.00348 0.99999 0.00119

0.01270 0.00069 0.99992

0510

Number of sources

152025

0

5

10

15

20

25

Error (Euclidean Distance)

Error of BGA-BSS Simulation

Error of CGA-BSS Simulation

Error of PSO-BSS Simulation

0510

Number of sources

152025

0

100

200

300

400

500

600

Time Consumption (Second)

Iterative time for BGA-BSS

Iterative time for CGA-BSS

Iterative time for PSO-BSS

Page 10

Research of Blind Signals Separation…

Figure 11-15 show the source signals that are two speech signals and a random noise

signal, mixed signals and the estimated signals based on CGA, BGA and PSO. The

kurtosises of sources are 8.585, 4.7775, and -1.1946 respectively. Table IV gives the

Euclidean distance and SNR values of estimated signals. The SNR values show that

sources and obtained signals for BSS problem based on CGA and PSO have the most

similarity. The effectiveness of these algorithms in reduction some of noises from

speech signals such as white noise, factory noise and babble noise is indicated in

experimental results. The error diagram of applying these evolutionary algorithms to

speech signals in the presence of noise is shown in Figure 16. It is deduced from the

results of separating based on PSO that estimated signals are separated with fast

convergence and high accuracy in less time than other algorithms.

Figure 10. The best fitness for BSS based on BGA,

CGA and PSO forsolving BSS problem

Figure 12. Mixed speech signals with random

noise

Figure 14. Separate signals y obtain from BGA

algorithm

S.Mavaddaty & A.Ebrahimzadeh.

86

Figure 11. Two speech signals and noise signals as

input sources

Figure 13. Separate signals y obtain from CGA

algorithm

Figure 15. Separate signals y obtain from PSO

algorithm

020 406080 100120140 160 180200

-170

-160

-150

-140

-130

-120

-110

-100

-90

-80

Generation

Fitness

Bst solution of BGA

Bst solution of CGA

Bst solution of PSO

02000400060008000100001200014000

-10

0

10

02000400060008000100001200014000

-10

0

10

02000400060008000100001200014000

-10

0

10

02000400060008000100001200014000

-10

0

10

02000400060008000100001200014000

-10

0

10

02000400060008000 100001200014000

-2

0

2

02000400060008000100001200014000

-2

0

2

0 20004000 60008000100001200014000

-10

0

10

0 200040006000 8000100001200014000

-10

0

10

0 2000 400060008000100001200014000

-10

0

10

0 2000 4000 60008000100001200014000

-2

0

2

0 2000 400060008000100001200014000

-10

0

10

0 20004000 6000 8000100001200014000

-10

0

10

0 2000 400060008000100001200014000

-10

0

10

0 200040006000 8000100001200014000

-2

0

2

Page 11

Journal of Computer Engineering

C. Separation of signals mixed with sub Gaussian signals

1 (2009) 77-88

87

In this simulation, the proposed algorithm is applied to signals with sub Gaussian

signals such as cosine wave that is probably mixed with original signals in

undesirable environmental conditions. The result is shown in Table V. Once again, it is

concluded that the proposed method based on CGA and PSO achieve the successful

separations of signals from their linear mixtures.

Figure 16. The error diagram with increasing the number of speech sources

in simulation of CGA, BGA and PSO for solving

BSS problem in the presence of random noise

Table 4. Comparison of Euclidean Distance criterion and SNR for three

algorithms in the presence of white noise

PSO Parameter

19.943

SNR of (

10.935

SNR of (

8.932

SNR of (

0.0046896 Euclidean Distance

Table 5. Comparison of Euclidean Distance criterion and SNR for three

algorithms for original signals mixed with cosine wave

PSO

Parameter

Conclusion

In this paper, the estimation of sources signals was executed using the evolutionary

mechanism of continuous and binary genetic algorithm and particle swarm optimization. The

proposed algorithm is based on mutual information and high order statistics of kurtosis. We

concluded that continuous genetic algorithm and particle swarm optimization are practically

effective without turbulence in converging to separate different number of source signals. But

CGA

19.424

11.497

8.877

0.0012222

BGA

13.423

18.072

7.345

0.0089968

1y )

2y )

3y )

CGA BGA

23.416 13.914 20.636

SNR of (

1y )

13.21 19.446 13.192

SNR of (

SNR of (

Euclidean Distance

2y )

8.732

0.0027212

7.987

0.011623

9.532

0.0012731

3y )

0510

Number of sources

152025

0

5

10

15

20

25

30

Error (Euclidean Distance)in the precense of noise

Error of BGA-BSS Simulation

Error of CGA-BSS Simulation

Error of PSO-BSS Simulation

Page 12

Research of Blind Signals Separation…

binary genetic algorithm has divergence and low accuracy. According to the proposed

simulations, in applications that need high speed and minimum cost of time computations, PSO

obtains better results than CGA and BGA for solving separation problems in multiple sources

signals. The experimental results indicated the effectiveness of this method in reducing some

noises such as random noise, white noise and babble noise from speech signals. Also, it showed

that there is no limitation on distribution of the original signals to enable the system to extract

up to three sources from the observed signals. The proposed methods overcome the local

minima problem occurred in the conventional gradient-based and neural network methods, and

yields global optimal solutions to linear blind source separation problems.

S.Mavaddaty & A.Ebrahimzadeh.

88

References

[1] M. Kadou, K. Arakawa, “A Method of Blind Source Separation for Mixed Voice Separation in

Noisy and Reverberating Environment”, IEICE Tech. Rep., vol. 108, no. 461, SIS2008-81, pp. 55-

59, March 2009.

[2] Z. Ding, Y. Li, “Blind Equalization and Identification”, Marcel Dekker, 2001.

[3] A. Hyvarinen, et al., “Independent Component Analysis” ,John Wiley & Sons Co, 2001.

[4] H. Yin and I. Hussain, “Blind Source Separation and Genetic Algorithm for Image Restoration”,

Advances in Space Technologies, 2006 International Conference, Issue, Page(s):167 – 172, Sept

2006.

[5] J.F. Cardoso, C.N.R.S, and E.N.S.T., “Blind signal separation: statistical principles”, Proceedings of

the IEEE, vol.86, NO 10, pp.2009-2025, OCT. 1998.

[6] M. Kuraya, U. Atsushi, Y Shigeru and K. Umeno “Blind source separation of chaotic laser signals

by independent component analysis”, Optics Express, Vol. 16, Issue 2, pp. 725-730, Jan 2008.

[7] Z. Shi, Z. Jiang and F. Zhou, “A fixed-point algorithm for blind source separation with nonlinear

autocorrelation,” Journal of Computational and Applied Mathematics, 223 908–915, 2009.

[8] J. LeBlanc and P. Leon, “speech separation by kurtosis maximization”, proc. ICASSP.vol.2, pp

1029-1032, 1998.

[9] S. Sun, J. Zheng and D. Wu, “Research on blind source separation based on natural gradient

algorithm”, journal of airforceering university (natural science edition), vol. 4, pp. 50-54, Jun 2003.

[10] S. Sun and J. Zheng, “Blind source separation of communication signals of different magnitudes”,

Journal of china institute of communications, vol 25, pp. 132-138, June 2004.

[11] Y. Tan, J. Wang, “Nonlinear blind source separation using higher order statistics and a genetic

algorithm,” IEEE Transactions on evolutionary computation, vol. 5, no. 6, Dec 2001.

[12] S. Kai, W. Qi and D. Mingli, “Approach to Nonlinear Blind Source Separation Based on Niche

Genetic Algorithm”, Proceedings of the Sixth Interrnational Conference on Intelligent Systems

Design and Applications, 2006.

[13] P. Zheng, Y. Liu, L. Tian, Y. Cao, “A Blind Source Separation Method Based on Diagonalization

of Correlation Matrices and Genetic Algorithm ” Fifth World Congress, Vol. 3, Issue, pp. 2127 –

2131, vol.3, June 2004.

[14] X. Y. Zeng, Y. W. Chen, Z. Nakao and G. Yamashita, “Signal separation by independent

component analy-sis based on a genetic algorithm”, 5th International Conference, Vol. 3, Issue,

vol.3, pp. 1688-1694, 2000.

[15] K. Wang, W. Zhang, “Blind Source Separation Based on Chaotic Immune Genetic Algorithm with

High order Cumulate”, IEEE International Conference, Volume, Issu, pp. 139 – 143, Dec. 2006.

[16] W. Yu, L. Zhenxing and L. Chinghai, “Improved Particle Swarm to Nonlinear Blind Source

Separation”, International Symposium on Microwave, Antenna, Propagation and EMC

Technologies for Wireless Communications, Aug 2007.

[17] Y. Gao, S. Xie, “A blind source separation algorithm using particle swarm optimization”,

Proceedings of the IEEE 6th Circuits and Systems Symposium, Vol. 1, Issue , pp. 297 - 300 Vol.1,

31 May-2 June 2004.

[18] F. Abrard, Y. Deville and J. Thomas, “Blind partial separation of underdetermineed convolutive

mixtures of complex sources based on differential normalized kurtosis”, Neurocomputing, pp

.2071–2086, 2008.

[19] M. Taoufikib, A. Adiba snd D. Aboutajdine, “Blind separation of any source distributions via

high-order statistics”, Signal Processing, pp. 1882–1889, 2007.

[20] A. K. Nandl, “Blind Estimation Using Higher Order Statistics,” Kluwer Academic Pub, 1999.

[21] S. sun, J. Zheng. “Blind source separation of communication signals of different magnitudes”

Journal of china institute of communications, vol 25, pp. 132-138B, June 2004.

[22] A. Hyvärinen and E. Oja, “Independent Component Analysis:Algorithms and Applications”,

Neural Networks, vol 13, pp. 411-430, 2000.