criterion for blind signals separation based on correlation function

摘 要: 盲信号 分离的依据 通常是信 号的独立性 或利用它 们的高阶累 积量。考虑了双通 道的盲分 离问题, 提出了一种基于相关函数的盲 分离准则。证 明了在源信号不相 关的条件下, 可以实现源信 号的盲分离。并给出了一种可直接求解二次方程组的盲分离算法。 关键词: 盲信号分离; 独立分量分析; 累积量; 相关函数 文章编号: 1000-9361 2003 03-01

Received dat e: 2002-10-22; R evision received dat e: 2003-05-20

A rt icle U RL: ht t p: / / ww w hkx b n et cn/ cja/ 2003/ 03/ 0162/

Criterion for Bl ind Signal s Separation Based on

Correlation Function SON G Y ou1, L IU Zhong -kan2

, L I Qi-han3

( 1 School of Sof tw are, Beij ing University of A eronautics and A stronautics, Beij ing　100083, China) ( 2 School of Science, Beij ing U niversity of A er onautics and A stronautics, B eij ing　100083, China)

( 3 Dep artment of Prop ulsion, Beij ing University of A eronautics and A stronautics,

Beij ing　100083, China)

Abstract: 　 Blind separ ation of sour ce signals usually r elies either o n the condition o f statistically

in-dependence or invo lving their higher -order cumulants T he model o f tw o channels sig nal separ ation is

consider ed A crit er ion based on cor relatio n functions is pr oposed I t is pr oved that the signals can be

separ ated, using only t he condition of no ncor relatio n A n alg orit hm is derived, w hich only involves

the solutio n to quadric nonlinear equations.

Key words: 　blind signals separation; independent compo nent analysis; cumulants; corr elation

func-tion

基于相关函数的盲信号分离准则 宋　友, 柳 重堪, 李其 汉 中国航空 学报( 英 文版) 2003, 16( 3) :

162- 168.

摘　要: 盲信号 分离的依据 通常是信 号的独立性 或利用它 们的高阶累 积量。考虑了双通 道的盲分

离问题, 提出了一种基于相关函数的盲 分离准则。证 明了在源信号不相 关的条件下, 可以实现源信

号的盲分离。并给出了一种可直接求解二次方程组的盲分离算法。

关键词: 盲信号分离; 独立分量分析; 累积量; 相关函数

文章编号: 1000-9361( 2003) 03-0162-07　　　中图分类号: V 243　　　文献标识码: A

　 　 T he problem of Blind Sources Separation

( BSS) arises in diverse fields of science and eng

i-neering like speech analysis and recognit ion, array

processing, m ultiuser detection, data com

munica-tion, imag e recovery, feat ure ex t ract ion, denoise,

etc Consequently, many w orks of BSS have been

present ed, including theories, algorit hms and

ap-plicat ions[ 1-13]

In this paper, t he tw o channels case is

consid-ered Mathemat ically , t he m odel is in brief

de-scribed by

x = As = x1( t)

x2( t) =

1 a12

a21 1

s1( t)

s2( t) ( 1)

v = Bx = v1( t)

v2( t) =

1 b12

b21 1

x1( t)

x2( t) = BAs = Cs = c11 c12

c21 c22

s1( t)

s2( t) ( 2)

w here x= [ x1( t ) 　x2( t ) ]Tare observed sig nals, s

= [ s1( t) 　s2( t ) ]Tare unknow n source signals, and coupling syst em A is the unknown const ant mat rix

or linear time invariant ( LT I) syst em T he xi( t ) result s f rom measurement s by sensors receiving contribut ions f rom sources, which is coupled by sources s1and s2 T he t ask of BSS is t o design a re-const ruction syst em B acting on x, w hich can elim-inat e t he coupling ef fects bet ween s1 and s2, and

w it h t he output signals v= [ v1　v2]T

, the compo-nent sig nal of w hich is a good estimate of sources

s, that is vi∝sj( i, j = 1, 2) In other w ords, t he purpose of BSS is to obt ain the separated system

C, which is in t he form of diagonalization ( or re-verse diag onalizat ion) as follow s

C = c11 0

0 c22

, or C = 0 c12

c21 0 ( 3)

Here, t he sources are said to be separable How

ev-er, since in m ost sit uat ions t he sources and t he

coupling syst em are unknow n, w hich m akes it not

av ailable to recover and ex t ract the usef ul signals

from sources T he signal separation is blind in t his

condit ion Wit h t he dif ficult y in theory and t he

im portance in application, the study of BSS is

valuable

Since t he 1990s, there has been a subst antial

development about the t heories and applicat ions

re-search for BSS T he main t heory rere-searches

in-clude charact eristics of sources, separat ion criteria,

separation condit ions, et c In addit ion, m ost

algo-rit hm s and applicat ions are also proposed A

fun-damental assum ption is that t he source signals are

stat ist ically independent , w hich has developed an

im portant separat e method called Independent

Com ponent A naly sis ( ICA )[ 4, 8, 12, 13] Mutual

infor-mation is a basic measurem ent of independence

be-tw een sources, and it can be represent ed by t he

sum of higher-order cum ulants in T aylor series

ex-pansion T hus t he separat ion condition can be

w eakened t o require only hig her-order st atistics

( cum ulants or multispectrums) As a result ,

an-other crucial criterion has been proposed, w hich is

based on higher-order statistics[ 5, 7, 11] T he

second-order statist ic includes t he uncorrelat ed inf ormat ion

betw een sources, and it has a clear meaning and

feasible comput abilit y , compared w it h

higher-or-der stat ist ics O n the assumpt ion of noncorrelat ion

betw een sources ( in f act , the assum ption of

non-correlat ion or independence is reasonable f or diff

er-ent sources ) , a novel met hod t hat implem er-ent s

minimizat ion of t he quadrat ic sum of correlat ion

functions betw een output signals v1 and v2, was

proposed[ 9, 10]

A new method f or BSS has been proposed in

this paper, w hich is based on a crit erion of

correla-tion funct ions being equal to zero bet ween out put

sig nals It is proved that noncorrelation is a suff

cient condit ion for signals separat ion A given eff

i-cient algorithm is derived f rom t he crit erion, w hich

only involves the direct solution t o quadric

nonlin-ear equat ions

1　BSS for Constant System

1 1　Separation criterion Consider the undet erm ined coupling syst em A

as a constant m at rix f irst ly

L et s1( n) and s2( n) be t im e series signals De-fine t heir correlat ion f unct ion by

Rs

1s2( m) = E{ s1( n) s2( n + m) } =

∑

n s1( n) s2( n + m) ( 4)

w here m belong s to integer s1( n) and s2( n) are said to be of noncorrelat ion if Rs

1s2( m) = 0, m Given s1 and s2 are incorrelat e Suppose t he sources are separable, t hat is t o say C is in t he form of Eq ( 3) T hen v1 and v2 are also incorre-lat e It can be obtained by

Rv

1v2( m) = E { v1( n) v2( n + m) } = E{ c11s1( n) c22s2( n + m) } =

c11c22E{ s1( n) s2( n + m) } =

c11c22Rs1s2( m) = 0 ( 5)

T hus, a necessary condition f or signal separat ion is

t hat v1and v2be incorrelate It can be proved that

Rv

1v2( m) = 0 is also a suff icient condition for signal separation

Theorem 1　 Given s1 and s2 are incorrelate,

Rsisi( m) is the autocorrelation function of si( i = 1, 2) , Suppose

Rsisi( m1) ≠ Rsisi( m2) , m1, m2, i = 1, 2 ( 6) det

Rs

1s1( m1) Rs

2s2( m1)

Rs

1s1( m2) Rs

2s2( m2) ≠ 0 ( 7) det C = det c11 c12

c21 c22

T hen C is in the form of Eq ( 3) , if

Rv

1v2( m) = 0 ( 9) 　　Proof　Rs

1s2( m) = 0 and Rv

1v2= 0, t hen

Rv

1v2( m) = E { v1( n) v2( n + m) } =

E { [ c11s1( n) + c11s2( n) ] × [ c21s1( n + m) + c22s2( n + m) ] } =

E { c11c21s1( n) s1( n + m) + c11c22s1( n) s2( n + m) +

c12c21s2( n) s1( n + m) + c12c22s2( n) s2( n + m) } =

c12c21Rs1s1( m) + c11c22Rs1s2( m) +

c12c21Rs

2s1( m) + c12c22Rs

2s2( m) =

c11c21Rs s( m) + c12c22Rs s( m) = 0

By Eq ( 6) , then

c11c21Rs

1s1( m1) + c12c22Rs

2s2( m1) = 0

c11c21Rs

1s1( m2) + c12c22Rs

2s2( m2) = 0

By Eq ( 7) , then 　 c11c21= c12c22= 0

T hus c11= 0 or c21= 0 , and c12= 0 or c22 = 0

By Eq ( 8) , then　 c11c22- c12c21≠ 0

If c11= 0 t hen c12≠0, c21≠0, c22= 0

If c21= 0 t hen c11≠0, c22≠0, c12= 0

If c22= 0 t hen c12≠0, c21≠0, c11= 0

If c12= 0 then c11≠0, c22≠0, c21= 0

Hence, C is in t he f orm of Eq ( 3)

1 2　Al gorithm

T o achieve the desired signals separation, one

can reformulate the problem as that of solving

bi-nary quadric nonlinear equat ions L et b1= b12 and

b2= b21for simplicit y A ccording t o the

noncorrela-tion bet w een v1and v2, one can obt ain

Rv

1v2( m) = E{ v1( n) v2( n + m) } =

　　E{ [ x1( n) + b1x2( n) ] ×

　　[ b2x1( n + m) + x2( n + m) ] } =

　　b2R11( m) + R12( m) + b1b2R21( m) +

　　b1R22( m) = 0

( 10)

w here Rij( m ) = Rxixj( m ) = E { xi( n) xj( n+ m) }

T hus, w it h m1≠m2, one can have

b2R11( m1) + R12( m1) + b1b2R21( m1) +

　　　b1R22( m1) = 0

b2R11( m2) + R12( m2) + b1b2R21( m2) +

　　　b1R22( m2) = 0

( 11)

It can be observed t hat t he undet ermined b1and b2

depend only on the correlation funct ions bet w een

the observed sig nals x1 and x2 By Eq ( 11) , one

can obt ain the equivalent equations

b2= - ( 3b1+ 2) / 1

b2+ b1+ != 0 ( 12)

w here

1= R11( m1) R21( m2) - R11( m2) R21( m1) ,

2= R12( m1) R21( m2) - R12( m2) R21( m1) ,

3= R21( m2) R22( m1) - R21( m1) R22( m2) ,

= 3

= R11( m2) R22( m1) - R11( m1) R22( m2) +

　　R12( m1) R21( m2) - R12( m2) R21( m1) ,

!= R11( m2) R12( m1) - R11( m1) R12( m2)

Eq ( 12) has tw o analyt ic solut ions, t hat may real-ize the blind separat ion One of t he solut ions is in response t o C f or diagonalization, while the ot her

is in response t o C for reverse diagonalization

It is also possible t o apply the num erical method f or solving the equat ions here

1 3　Experiment results

T he algorit hm w as t est ed in t he f ollow ing sce-nario: T he sources s1 and s2 w ere speech signals from a man and a w oman respect ively A ssume t he sources are noncorrelation ( by calculat ion one can have Rs

1s2( m ) ≤2×10- 3, so the assum ption is reasonable ) T he mix ing m at rix w as chosen at random as

- 0 37 1 Implement ing t he algorit hm yields t he decoupled matrix B, w hich can eliminate t he coupling eff ect s betw een s1and s2f rom observed signals x1and x2 Figs 1 and 2 show the sources F ig s 3 and 4 rep-resent t he observed sig nals F ig s 5 and 6 depict

t he out put signals, respectively

F ig 1　So ur ce signal s 1 ( n)

F ig 2　So ur ce signal s 2 ( n)

Fig 3　Observed sig nal x 1 ( n)

T hrough t he calculation, one can obt ain

F ig 4　Observed sig nal x 2 ( n)

Fig 5　Output signal v 1 ( n)

Fig 6　Output signal v 2 ( n)

C = 1 2560 - 0 0720

0 0093 1 2351 Obviously, C has the approx imat e form of a

diago-nal matrix T hen

v1( n) ≈ s1( n) , v2( n) ≈ s2( n)

Simulat ion result verif ies the validit y of t he

pro-posed method

In fact , t he assum ption t hat t he diag onal

en-tries of A equal to 1 as Eq ( 1) is not necessary

N ow , apply the algorithm to separat ion bet w een a

vibrat ion signal and a G auss noise w it h the random

mixing matrix

A = 0 8351 - 0 2193

0 5287 0 9219

By implement ing the algorit hm, one can have

- 0 6345 1

C = 0 9608 - 0 0000

- 0 0011 1 0611 Clearly, C is almost diagonal, w hich can eliminat e

the coupling eff ects eff icient ly and separat e t he

mixing sources successf ully

2　BSS for L T I System

2 1　Separation criterion Consider the more general case in w hich t he coupling syst em A is an unknow n L T I syst em R e-ferring t o Eq ( 1) , the frequency response of A is

A( ∀) = 1 A12( ∀)

L et t he coupling syst em ( filt ers) be represent ed in

t he Z-domain

A12( z ) = ∑

q

i= 0

a12( i) z- i

A21( z ) = ∑

q

i= 0

a21( i) z- i

( 14)

Here, t he task of BSS is to design a reconst ruction

L T I sy st em B so that separated sy stem C is in t he form of Eq ( 3) R ef erring to Eq ( 14) , represent

B in the Z-domain as follow s

B12( z ) = ∑

r

1

i= 0

b12( i) z- i

B21( z ) = ∑

r

2

i= 0

b21( i) z- i

( 15)

　 　Assum e that s1 and s2 are incorrelat e T hen

t he noncorrelat ion bet ween v1and v2is also a suff i-cient condition f or signal separation of the L T I sys-tem

For simplicity , consider t he coupling filt ers be first -order, viz q1= q2= 1 T o implement separa-tion, let t he decoupling syst em be also first-order, viz r1= r2= 1 T hen Eq ( 1) and Eq ( 2) become

x1( n) = s1( n) + a12( 0) s2( n) + a12( 1) s2( n - 1)

x2( n) = s2( n) + a21( 0) s1( n) + a21( 1) s1( n - 1)

( 16)

v1( n) = x1( n) + b12( 0) x2( n) +

　　　b12( 1) x2( n - 1)

v2( n) = x2( n) + b21( 0) x1( n) +

　　　b21( 1) x1( n - 1)

( 17)

Subst it ut ing Eq ( 16) int o Eq ( 17) gives

v1( n) = ∑2

i= 0

c11( i) s1( n - i) + ∑1

j = 0

c12( j ) s2( n - j )

v2( n) = ∑1

i= 0

c21( i) s1( n - i) + ∑2

j = 0

c22( j ) s2( n - j )

( 18)

w here

c11( 0) = 1+ b12( 0) a21( 0) ,

c11( 1) = b12( 0) a21( 1) + b12( 1) a21( 0) ,

c11( 2) = b12( 1) a21( 1) ,

c12( 0) = a12( 0) + b12( 0) ,

c12( 1) = a12( 1) + b12( 1) ,

c21( 0) = a21( 0) + b21( 0) ,

c21( 1) = a21( 1) + b21( 1) ,

c22( 0) = 1+ b21( 0) a12( 0) ,

c22( 1) = b21( 0) a12( 1) + b21( 1) a12( 0) ,

c22( 2) = b21( 1) a12( 1)

T he correlat ion f unct ions betw een v1and v2are

Rv

1v2( m) = E{ v1( n) v2( n + m) } =

　　c11( 0) c21( 1) R1( m - 1) +

　　[ c11( 0) c21( 0) + c11( 1) c21( 1) ] R1( m) +

　　[ c11( 1) c21( 0) + c11( 2) c21( 1) ]

　　R1( m + 1) + c11( 2) c21( 0) R1( m + 2) +

　　c12( 0) c22( 2) R2( m - 2) +

　　[ c12( 0) c22( 1) + c12( 1) c22( 2) ] R2( m - 1) +

　　[ c12( 0) c22( 0) + c12( 1) c22( 1) ]

　　R2( m) + c12( 1) c22( 0) + R2( m + 1)

( 19)

w here R1= Rs

1s1, R2= Rs

2s2 If v1and v2are incorre-late, then Rv

1v2( m) = 0, m Hence

Rv

1v2( m1) = 0 　

Rv

1v2( m8) = 0

( 20)

T he coef ficient mat rix R of Eq ( 20) is

R 1 ( m 1 - 1) …R 1 ( m 1 + 2) R 2 ( m 1 - 2) …R 2 ( m 1 + 1)

R 1 ( m 8 - 1) …R 1 ( m 8 + 2) R 2 ( m 8 - 2) …R 2 ( m 8 + 1)

If

the equations ( 20) have only zero solutions as f

ol-low s

c11( 0) c21( 1) = 0,

c11( 0) c21( 0) + c11( 1) c21( 1) = 0,

c11( 1) c21( 0) + c11( 2) c21( 1) = 0,

! c11( 2) c21( 0) = 0,

∀ c12( 0) c22( 2) = 0,

# c12( 0) c22( 1) + c12( 1) c22( 2) = 0,

∃ c12( 0) c22( 0) + c12( 1) c22( 1) = 0,

% c12( 1) c22( 0) = 0

T he Z-t ransf orm of Eq ( 18) is

V1( z ) = C11( z ) S1( z ) + C12( z ) S2( z )

V2( z ) = C21( z ) S1( z ) + C22( z ) S2( z ) ( 22)

T he separat ed sy stem C represent ed in Z-domain is

C( z ) = C11( z ) C12( z )

C21( z ) C22( z ) ( 23)

w here

C11( z ) = c11( 0) + c11( 1) e- z + c11( 2) e- 2z,

C12( z ) = c12( 0) + c12( 1) e- z,

C21( z ) = c21( 0) + c21( 1) e- z,

C22( z ) = c22( 0) + c22( 1) e- z + c22( 2) e- 2z Suppose t hat

detC( z ) ≠ 0, z ( 24) and t hat the condit ion of Eq ( 21) is sat isf ied

T hus, t he sources are separable according t o t he condit ion of noncorrelat ion Since those conditions are sat isf ied, t he equat ions ( 20) have only zero so-lutions as -% T he eig ht solut ions w ill, in t he sequel, be analyzed as f ollow s

T hese analy ses st art from and !, thus

c11( 0) = 0 　or　c21( 1) = 0, and

c11( 2) = 0 　or 　c21( 0) = 0

T here are four cases A nalyze them respect ively ( 1) If c11( 0) = 0 and c11( 2) = 0, then by and one can obt ain c11( 1) c21( 1) = 0 and c11( 1)

c21( 0) = 0

If c11( 1) = 0, t hen ( i) by Eq ( 24) , one can obtain c12( 0) ≠0 or

c12( 1) ≠0, and c21( 0) ≠0 or c22( 1) ≠0 If c12( 0)

≠ 0 and c12( 1) ≠0, then by ∀ c22( 2) = 0, by %

c22( 0) = 0, and by # c22( 1) = 0; If c12( 0) = 0 and

c12( 1) ≠0, then by % c22( 0) = 0, by ∃ c22( 1) =

0, and by # c22( 2) = 0; If c12( 0) ≠0 and c12( 1) =

0, then by ∃ , # and ∀ c22( 0) = 0, c22( 1) = 0 and c22( 2) = 0 respectively T hus, C( z ) is in the form of reverse diagonalization

If c11( 1) ≠0, t hen ( ii) c21( 0) = 0 and c21( 1) = 0, by Eq ( 24) one can obt ain that at least one of c22( 0) , c22( 1) and c22( 2) is not equal t o zero If c22( 0) ≠0, then

by % c12( 1) = 0, and by ∃ c12( 0) = 0; If c22( 2) ≠

0, t hen by ∀ c12( 0) = 0, and by # c12( 1) = 0; If

c22( 1) ≠0 and c22( 0) = c22( 2) = 0, then by # c12

( 0) = 0, and by ∃ c12( 1) = 0 T hus, C( z ) is in

the form of diag onalizat ion

( 2) If c21( 0) = c21( 1) = 0, it is t he same as

( ii)

( 3) If c11( 0) = 0 and c21( 0) = 0, then by

and c11( 1) c21( 1) = 0 and c11( 2) c21( 1) = 0 If c21

( 1) = 0, it is t he sam e as ( ii) ; If c21( 1) ≠0 then

c11( 1) = 0 and c11( 2) = 0, it is t he same as ( i)

( 4) If c21( 1) = c11( 2) = 0, t hen by and

c11( 0) c21( 0) = 0, c11( 1) c21( 0) = 0 If c21( 0) = 0, it

is t he sam e as ( ii) ; If c21( 0) ≠0, then c11( 0) = 0

and c11( 1) = 0, it is t he same as ( i)

When the analyses st art from ∀ and % , t hey

are t he same as ( 1) ～( 4)

In fact , t he situat ion t hat C( z) is in the form

of reverse diag onalization can not occur when t he

filter channels are f irst order ( or any finite order)

It is because

C( z ) = B( z ) A( z) =

1 B12( z)

B21( z ) 1

1 A12( z )

A21( z ) 1 here one can have

C11( z ) = [ 1 + b12( 0) a21( 0) ] + [ b12( 0) a21( 1) +

b12( 1) a2 1( 0) ] e- z + b12( 1) a21( 1) e- 2z

C22( z ) = [ 1 + b21( 0) a12( 0) ] + [ b21( 0) a12( 1) +

b21( 1) a1 2( 0) ] e- z + b21( 1) a12( 1) e- 2z

For first order filt er channels, b12( 1) ≠0, b21( 1) ≠

0, a12( 1) ≠0, a21( 1) ≠0 Hence, C11( z ) ≠0, C22( z )

≠0

In conclusion, t o achieve sources separat ion in

the case of f irst order filter channels, it is

neces-sary t o require t hat t he condit ions in Eq ( 21) and

Eq ( 24) are satisfied In the case of high order f

il-ters ( great er than one) , one may guess t hat t he

BSS can be realized by using only the correlat ion

functions, provided that t he conditions parallel to

Eq ( 21) and Eq ( 24) are sat isf ied It can be seen

that t he case of f irst order L T I is more complex

than the case of const ant mat rix sy stem When t he

order of filt ers is great er t han one, the terms need

to be analy zed further

2 2　Al gorithm

T o achieve the decoupling coef ficients b12( 0) ,

b12( 1) , b21( 0) and b21( 1) as Eq ( 15) For

sim-plicit y, L et [ b1, b2, b3, b4] = [ b12( 0) , b12( 1) , b21

( 0) , b21( 1) ] A ccording to the noncorrelat ion

be-tw een v1and v2, one can obt ain

Rv1v2( m) = E{ v1( n) v2( n + m) } = 　　E{ [ x1( n) + b1x2( n) + b2x2( n - 1) ] × 　　[ x2( n + m) + b3x1( n + m) +

　　b4x1( n + m - 1) ] } =

　　R21( m) b1b3 + R21( m - 1) b1b4+

　　R21( m + 1) b2b3+ R21( m) b2b4+

　　R22( m) b1+ R22( m + 1) b2+

　　R11( m) b3+ R11( m - 1) b4+ R12( m) = 0

( 25)

w here Rij( m ) = Rxixj( m) T hus, w it h t he num-bers m1≠m2≠m3≠m4, one can obtain the quadric equat ions, w here the unknown variables are [ b1,

b2, b3, b4] Som e numerical m et hods m ay be ap-plied f or solving t he equat ions t o realize signals sep-aration

2 3　Experiment results

T he algorit hm w as t est ed in t he f ollow ing sce-nario: T he sources s1and s2w ere the same speech sig nals as those in Section 1 3 T he coupling coef-ficient s corresponding to Eq ( 14) w ere chosen as [ a12( 0) , a12( 1) ] = [ 0 71, - 0 53] , [ a21( 0) , a21( 1) ] = [ 0 12, 0 37] From Eq ( 18) one can know t hat t he theoret ical decoupling coef ficients corresponding t o Eq ( 15) are

[ b12( 0) , b12( 1) ] = - [ a12( 0) , a12( 1) ] =

[ - 0 71, 0 53] , [ b21( 0) , b21( 1) ] = - [ a21( 0) , a21( 1) ] =

[ - 0 12, - 0 37]

By implement ing t he algorit hm in Sect ion 2 2, one can obt ain the numerical decoupling coeff icient s as [ b12( 0) , b12( 1) ] = [ - 0 6919, 0 5019] , [ b21( 0) , b21( 1) ] = [ - 0 1325, - 0 3668]

T he absolut e errors of t hese coeff icient s bet w een

t heoretical and numerical values are g iven as T able 1

　　T he experim ent results indicate that num erical decoupling coef ficient s are close to t heoret ical val-ues

Tabl e 1　Errors between theoretical and

numerical val ues

D ecoupling Coef f icient s A bsolut e Errors

3　Conclusions

T he BSS problem of the tw o channels model

is considered in t his paper, w hich is based on

cor-relat ion funct ions It is proved that t he blind

sources can be separat ed using the condit ion that

they are incorrelate By im posing the condit ion on

the reconstruct ed sig nals, a criterion is obt ained

for signal separat ion, and an ef ficient alg orit hm is

given that only involv es the direct solut ion to

quadric nonlinear equations

References [ 1] 　 Jut ten C , Herault J Blind separat ion of sources, Part I:

A n adapt ive algorit hm based on neuromimet ic architect ure

[ J] Signal Processing, 1991, 24( 1) : 1- 10.

[ 2] 　 Comon P, J utt en C, Herault J Blind separat ion of sources,

Part II: Problem st at emen t [ J] Signal Processing, 1991,

24( 1) : 11- 20.

[ 3] 　 Sorluchyari E, Jut ten C, Herault J Blind separat ion of

sources, Part III: St abilit y analysis [ J ] Signal Processing,

1991, 24( 1) : 21- 29.

[ 4] 　 Comon P Independen t component analysis, a new concept

[ J] Signal Processing, 1994, 36: 287- 314.

[ 5] 　 Y ellin D , W einst ein E Criteria for multich annel signal sep-arat ion[ J] IEE E T rans on Signal Processin g, 1994, 42: 2158- 2168.

[ 6] 　 Th i H L N , Jut ten C Blind source separation f or convolu-tive mixt ure[ J] Signal Processing, 1995, 45: 209- 229 [ 7] 　 Y ellin D , W einst ein E M ultich annel signal separation:

M et hods an d analysis[ J] IEE E T rans on Signal Process-ing, 1996, 44: 106- 118.

[ 8] 　 Cichochi A , U nbehauen R R obust neural net w orks w it h

on -lin e learning for blind ident if icat ion and blind separation

of sources [ J ] IEEE T rans on Circuit s an d Syst ems-I,

1996, 43( 11) : 894- 906.

[ 9] 　 Lindgren U A , Broman H S ource separation using a crit e-rion based on secon d-order st at ist ics [ J ] IEEE T rans on Signal Processing, 1998, 46( 7) : 1837- 1850.

[ 10] 　Sah lin H, Br oman H S eparation of real -w orld signals [ J ] Signal Processing, 1998, 64: 103- 113.

[ 11] 　Ihm B C, Park D J Blind separation of sources using high-er-order cum ulant s[ J] Signal Processing, 1999, 73: 267-276.

[ 12] 　Hyva!rinen A , O ja E Independent component analysis: al-gorit hms and applicat ions[ J] N eural Netw orks, 2000, 13: 411- 430.

[ 13] 　Cruces-A lvarez S , Cich ocki A , Cast edo-R ibas L A n itera-tive inversion approach to blind source separat ion[ J] IEEE Trans on N eural N et w ork s, 2000, 11( 6) : 1423- 1437. Biographies:

SONG You　Bor n in 1973, he r eceiv ed

B S fr om Beijing U niver sity of Aer o-nautics and A str oo-nautics in 1997 He received his doctoral degr ee in 2003, and then became a teacher there E-mail: song you@ buaa edu cn