V ol 16 N o A ugust 2003 CHIN ESE JO U RN A L OF A ERO N A U TICS Criterion for Bl ind Signal s Separation Based on Correlation Function SONG You , L IU Zhong -kan , L I Qi-han ( School of Sof tw are, Beij ing University of A eronautics and A stronautics, Beij ing 100083, China) ( School of Science, Beij ing University of A er onautics and A stronautics, B eij ing 100083, China) ( 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 , , ( ) 2003, 16( 3) : 162- 168 : , , : ; ; ; : 1000-9361( 2003) 03-0162-07 : V 243 T he problem of Blind Sources Separat ion ( BSS) arises in diverse fields of science and eng ineering like speech analysis and recognit ion, array processing, m ultiuser detection, data com munication, 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[ 1-13] plicat ions In this paper, t he tw o channels case is considered Mathemat ically , t he m odel is in brief described by x = As = v = Bx = x ( t) x ( t) v 1( t) v 2( t) BAs = Cs = = = s 1( t) a12 a21 s 2( t) b12 x ( t) b 21 x ( t) c11 c12 s1 ( t) c21 c22 s2 ( t) ( 1) = ( 2) 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/ :A x ( t ) ] T are observed sig nals, s w here x= [ x 1( t ) = [ s 1( t) T s 2( t ) ] are 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 x i ( t ) result s f rom measurement s by sensors receiving contribut ions f rom sources, which is coupled by sources s and s2 T he t ask of BSS is t o design a reconst ruction syst em B acting on x, w hich can eliminat e t he coupling ef fects bet ween s and s2 , and T w it h t he output signals v = [ v v ] , t he component sig nal of w hich is a good estimate of sources s, that is v i 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 reverse diag onalizat ion ) as follow s C= c 11 0 c22 , or C = c12 c21 ( 3) Crit erion f or Blind S ignals Separat ion Based on Correlat ion Funct ion A ugust 2003 Here, t he sources are said to be separable How ever , 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 research for BSS T he main t heory researches include charact eristics of sources , separat ion criteria , separation condit ions, et c In addit ion , m ost algorit hm s and applicat ions are also proposed A fundamental 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 Analy sis ( ICA) [ 4, 8, 12, 13] BSS for Constant System 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 s ( n) be t im e series signals Define t heir correlat ion f unct ion by R s1 s2 ( m) = E{ s ( n) s2 ( n + m) } = s1 ( n) s 2( n + m) sum of higher-order cum ulants in T aylor series expansion T hus t he separat ion condition can be w eakened t o require only hig her -order st atistics ( cum ulants or mult ispect rums ) As a result , another crucial criterion has been proposed, w hich is [ 5, 7, 11] T he second- w here m belong s to integer s1 ( n) and s2 ( n) are said to be of noncorrelat ion if R s1 s2 ( m ) = 0, m Given s1 and s 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 v and v are also incorrelat e It can be obtained by R v1 v ( m) = E { v 1( n) v 2( n + m) } = E{ c 11s1 ( n) c22 s2 ( n + m) } = c11c 22E{ s 1( n) s2 ( n + m) } = c11 c22R s1 s2 ( m) = R v v ( m) = is also a suff icient condition for signal separation Theorem Given s and s2 are incorrelate, R si si ( m ) is t he autocorrelation function of s i ( i = 1, 2) , Suppose R sisi ( m 1) feasible comput abilit y , compared w it h higher-order stat ist ics On the assumpt ion of noncorrelat ion det betw een sources ( in f act , the assum ption of noncorrelat ion or independence is reasonable f or diff erminimizat ion of t he quadrat ic sum of correlat ion functions betw een output signals v and v 2, 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- ( 5) T hus, a necessary condition f or signal separat ion is t hat v and v be incorrelate It can be proved that order statist ic includes t he uncorrelat ed inf ormat ion betw een sources, and it has a clear meaning and ent sources ) , a novel met hod t hat implem ent s ( 4) n Mutual infor- mation is a basic measurem ent of independence betw een sources, and it can be represent ed by t he based on higher-order statistics 163 R s i s i ( m 2) , m 1, m , i = 1, ( 6) R s s ( m 1) R s2 s2 ( m ) R s s ( m 2) R s2 s2 ( m ) det C = det c11 c12 c21 c22 0 T hen C is in the form of Eq ( 3) , if R v v ( m) = Proof ( 7) ( 8) ( 9) R s1 s2 ( m) = and R v v = 0, t hen R v1 v ( m) = E { v 1( n) v 2( n + m) } = E { [ c11s 1( n) + c 11s2 ( n) ] [ c21s 1( n + m) + c22s 2( n + m) ] } = tion funct ions being equal to zero bet ween out put sig nals It is proved that noncorrelation is a suff i- E { c11c21 s1 ( n) s1 ( n + m) + c 11c22s 1( n) s2 ( n + m) + cient condit ion for signals separat ion A given eff icient algorithm is derived f rom t he crit erion , w hich c12 c21R s1 s1 ( m) + c11 c22R s1 s2 ( m) + only involves the direct solution t o quadric nonlinear equat ions c12c21 s2 ( n) s 1( n + m) + c12c22s 2( n) s2 ( n + m) } = c12 c21R s2 s1 ( m) + c12 c22R s2 s2 ( m) = c11 c21R s1 s1 ( m) + c12 c22R s2 s2 ( m) = S ON G Y ou, LIU Zh ong -kan, LI Q i-han 164 CJA By Eq ( 6) , then c11c21 R s1 s1 ( m ) + c 12c22R s2 s2 ( m 1) = Eq ( 12) has tw o analyt ic solut ions, t hat may realize the blind separat ion One of t he solut ions is in c11c21 R s1 s1 ( m ) + c 12c22R s2 s2 ( m 2) = response t o C f or diagonalization, while the ot her By Eq ( 7) , then c11 c21 = c12c22 = T hus c11 = or c21 = , and c12 = or c22 = By Eq ( 8) , then c11c22 - c12 c21 11 12 21 22 If c = t hen c 0, c 0, c = If c21= t hen c11 0, c22 0, c 12= If c22= t hen c12 0, c21 0, c 11= If c12= t hen c11 0, c22 0, c 21= is in response t o C for reverse diagonalization Hence, C is in t he f orm of Eq ( 3) Al gorithm sources are noncorrelation ( by calculat ion one can 10 - 3, so the assum ption is have R s1 s2 ( m ) T o achieve the desired signals separation, one can reformulate the problem as that of solving binary quadric nonlinear equat ions L et b = b 12 and b2 = b 21 for simplicit y According t o the noncorrelation bet w een v and v 2, one can obt ain R v 1v ( m) = E{ v ( n) v 2( n + m) } = E{ [ x ( n) + b1 x 2( n) ] It is also possible t o apply the num erical method f or solving the equat ions here Experiment results T he algorit hm w as t est ed in t he f ollow ing scenario : T he sources s and s2 w ere speech signals from a man and a w oman respect ively Assume t he reasonable ) T he mix ing m at rix w as chosen at random as A= - 37 62 Implement ing t he algorit hm yields t he decoupled matrix B, w hich can eliminate t he coupling eff ect s b2R 11( m) + R 12 ( m) + b1b 2R 21 ( m) + betw een s1 and s f rom observed signals x and x Figs and show the sources F ig s and represent t he observed sig nals F ig s and depict b1R 22( m) = t he out put signals, respectively [ b2 x 1( n + m) + x 2( n + m) ] } = ( 10) w here R ij ( m ) = R x i x j ( m ) = E { x i ( n) x j ( n+ m) } T hus, w it h m m 2, one can have b2 R 11( m ) + R 12( m ) + b1 b2R 21( m ) + b 1R 22 ( m 1) = b2 R 11( m ) + R 12( m ) + b1 b2R 21( m ) + b 1R 22 ( m 2) = F ig So ur ce signal s1 ( n) F ig So ur ce signal s2 ( n) ( 11) It can be observed t hat t he undet ermined b1 and b depend only on the correlation funct ions bet w een the observed sig nals x and x By Eq ( 11) , one can obt ain the equivalent equations b = - ( 3b + 2) / ( 12) b + b + != w here = R 11( m ) R 21 ( m 2) - R 11 ( m 2) R 21 ( m 1) , = R 12( m ) R 21 ( m 2) - R 12 ( m 2) R 21 ( m 1) , = R 21 ( m 2) R 22 ( m 1) - R 21 ( m 1) R 22 ( m 2) , = = R 11( m2 ) R 22 ( m 1) - R 11 ( m 1) R 22 ( m 2) + R 12 ( m 1) R 21( m ) - R 12( m ) R 21( m ) , != R 11 ( m 2) R 12( m ) - R 11( m ) R 12( m ) Fig Observed sig nal x ( n) T hrough t he calculation, one can obt ain Crit erion f or Blind S ignals Separat ion Based on Correlat ion Funct ion A ugust 2003 165 mixing sources successf ully BSS for L T I System 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 ReF ig Observed sig nal x ( n) ferring t o Eq ( 1) , the frequency response of A is A 12 ( ∀) A( ∀) = ( 13) A 21 ( ∀) L et t he coupling syst em ( filt ers) be represent ed in t he Z -domain q A 12( z ) = a12 ( i) z - i a21 ( i) z - i i= ( 14) q Fig Output signal v ( n) A 21( z ) = i= 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) Ref erring to Eq ( 14) , represent B in the Z-domain as follow s r Fig Output signal v ( n ) B= C= 3793 - 6919 1 2560 - 0720 0093 2351 Obviously, C has the approx imat e form of a diagonal matrix T hen v 1( n) s1 ( n) , v 2( n) s 2( n) Simulat ion result verif ies the validit y of t he pro- b 12( i) z - i B 12( z ) = i= r ( 15) B 21( z ) = b 21( i) z - i i= Assum e that s1 and s are incorrelat e T hen t he noncorrelat ion bet ween v and v is also a suff icient condition f or signal separation of the L T I system For simplicity , consider t he coupling filt ers be first -order, viz q = q 2= T o implement separa- In fact , t he assum ption t hat t he diag onal entries of A equal to as Eq ( 1) is not necessary tion, let t he decoupling syst em be also first-order, viz r 1= r = T hen Eq ( 1) and Eq ( 2) become x ( n) = s1 ( n) + a12 ( 0) s2 ( n) + a12( 1) s2 ( n - 1) Now , apply the algorithm to separat ion bet w een a x ( n) = s2 ( n) + a21 ( 0) s1 ( n) + a21( 1) s1 ( n - 1) posed method ( 16) vibrat ion signal and a Gauss noise w it h the random v 1( n) = x ( n) + b12 ( 0) x ( n) + mixing matrix A= 8351 - 2193 9219 By implement ing the algorit hm , one can have 2379 B= - 6345 C= b12 ( 1) x 2( n - 1) 5287 9608 - 0000 - 0011 0611 Clearly, C is almost diagonal, w hich can eliminat e the coupling eff ects eff icient ly and separat e t he v 2( n) = x ( n) + b21 ( 0) x ( n) + ( 17) b21 ( 1) x 1( n - 1) Subst it ut ing Eq ( 16) int o Eq ( 17) gives v ( n) = c11( i) s 1( n - i) + c12( j ) s 2( n - j ) i= j= v ( n) = c21( i) s 1( n - i) + i= c22( j ) s 2( n - j ) j= ( 18) S ON G Y ou, LIU Zh ong -kan, LI Q i-han 166 w here CJA T he Z-t ransf orm of Eq ( 18) is V 1( z ) = C 11( z ) S ( z ) + C 12 ( z ) S 2( z ) c11 ( 0) = 1+ b12 ( 0) a21 ( 0) , c11 ( 1) = b12 ( 0) a21 ( 1) + b 12( 1) a21( 0) , ( 22) V 2( z ) = C 21( z ) S ( z ) + C 22 ( z ) S 2( z ) T he separat ed sy stem C represent ed in Z-domain is c11 ( 2) = b12 ( 1) a21 ( 1) , c12 ( 0) = a12( 0) + b 12( 0) , C( z ) = c12 ( 1) = a12( 1) + b 12( 1) , c21 ( 0) = a21( 0) + b 21( 0) , C 11( z ) C 12( z ) C 21( z ) C 22( z ) ( 23) w here - z c21 ( 1) = a21( 1) + b 21( 1) , C 11( z ) = c11 ( 0) + c11( 1) e + c11 ( 2) e - 2z , - z c22 ( 0) = 1+ b21 ( 0) a12 ( 0) , C 12( z ) = c12 ( 0) + c12( 1) e , c22 ( 1) = b21 ( 0) a12 ( 1) + b 21( 1) a12( 0) , C 21( z ) = c21 ( 0) + c21( 1) e , c22 ( 2) = b21 ( 1) a12 ( 1) C 22( z ) = c22 ( 0) + c22( 1) e- z + c22 ( 2) e - 2z - z T he correlat ion f unct ions betw een v and v are R v v2 ( m) = E{ v ( n) v ( n + m) } = Suppose t hat detC( z ) 0, ( 24) z c11( 0) c21( 1) R ( m - 1) + and t hat the condit ion of Eq ( 21) is sat isf ied [ c11( 0) c21 ( 0) + c11( 1) c21 ( 1) ] R ( m) + T hus, t he sources are separable according t o t he [ c11( 1) c21 ( 0) + c11( 2) c21 ( 1) ] R 1( m + 1) + c11 ( 2) c21 ( 0) R 1( m + 2) + condit ion of noncorrelat ion Since those conditions are sat isf ied , t he equat ions ( 20) have only zero so- c12( 0) c22( 2) R ( m - 2) + lutions as [ c12( 0) c22 ( 1) + c12( 1) c22 ( 2) ] R ( m - 1) + sequel, be analyzed as f ollow s T hese analy ses st art from [ c12( 0) c22 ( 0) + c12( 1) c22 ( 1) ] c11 ( 0) = R 2( m) + c12 ( 1) c22 ( 0) + R ( m + 1) w here R 1= R s1 s1 , R 2= R s2 s2 If v and v are incorrelate , then R v v ( m ) = 0, m Hence R v 1v 2( m1 ) = ( 20) or c11( 2) = ( 19) and !, thus c21( 1) = 0, and or c21 ( 0) = T here are four cases Analyze them respect ively ( 1) If c11 ( 0) = and c11 ( 2) = 0, then by and one can obt ain c11( 1) c21( 1) = and c11( 1) c21 ( 0) = If c11( 1) = 0, t hen R v 1v 2( m8 ) = ( i) by Eq ( 24) , one can obtain c 12( 0) T he coef ficient mat rix R of Eq ( 20) is R ( m1 - 1) R ( m + 2) R ( m - 2) R ( m + 1) R ( m8 - 1) R ( m + 2) R ( m - 2) R ( m + 1) c12 ( 1) ( 21) the equations ( 20) have only zero solutions as f ollow 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, 0, and c21 ( 0) or c22 ( 1) or If c12 ( 0) and c12 ( 1) 0, then by ∀ c22 ( 2) = 0, by % c22 ( 0) = 0, and by # c 22( 1) = 0; If c12( 0) = and c12( 1) If det R -% T he eig ht solut ions w ill, in t he 0, then by % c22 ( 0) = 0, by ∃ c22 ( 1) = 0, and by # c22( 2) = 0; If c12 ( 0) and c 12( 1) = 0, t hen by ∃ , # and ∀ c22 ( 0) = 0, c22 ( 1) = and c22 ( 2) = respectively T hus, C( z ) is in t he form of reverse diagonalization If c11( 1) 0, t hen ( ii) c21 ( 0) = and c21 ( 1) = 0, by Eq ( 24) one can obt ain that at least one of c22 ( 0) , c22 ( 1) ∀ c12( 0) c22 ( 2) = 0, # c12( 0) c22 ( 1) + c12( 1) c22( 2) = 0, ∃ c12( 0) c22 ( 0) + c12( 1) c22( 1) = 0, and c22( 2) is not equal t o zero If c22 ( 0) % c12( 1) c22 ( 0) = c22( 1) 0, then by % c12( 1) = 0, and by ∃ c12 ( 0) = 0; If c 22( 2) 0, t hen by ∀ c12( 0) = 0, and by # c12 ( 1) = 0; If and c22 ( 0) = c 22 ( 2) = 0, then by # c12 Crit erion f or Blind S ignals Separat ion Based on Correlat ion Funct ion A ugust 2003 ( 0) = 0, and by ∃ c12 ( 1) = 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) 167 plicit y, L et [ b 1, b2 , b 3, b4 ] = [ b 12 ( 0) , b 12 ( 1) , b 21 ( 0) , b21 ( 1) ] According to the noncorrelat ion betw een v and v 2, one can obt ain R v v ( m) = E{ v 1( n) v 2( n + m) } = ( 3) If c11 ( 0) = and c21 ( 0) = 0, then by E{ [ x 1( n) + b 1x ( n) + b2 x 2( n - 1) ] and c11( 1) c21( 1) = and c11( 2) c21( 1) = If c21 ( 1) = 0, it is t he sam e as ( ii) ; If c21 ( 1) then [ x 2( n + m) + b3x 1( n + m) + b 4x ( n + m - 1) ] } = c11( 1) = and c11 ( 2) = 0, it is t he same as ( i) R 21 ( m) b 1b3 + R 21( m - 1) b 1b + ( 4) If c21( 1) = c11 ( 2) = 0, t hen by and c11( 0) c21 ( 0) = 0, c11( 1) c21( 0) = If c21( 0) = 0, it is t he sam e as ( ii) ; If c21 ( 0) R 21 ( m + 1) b 2b3 + R 21( m) b 2b + R 22 ( m) b + R 22( m + 1) b + 0, then c11 ( 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) = R 11 ( m) b + R 11( m - 1) b + R 12 ( m) = ( 25) w here R ij ( m ) = R x i x j ( m) T hus, w it h t he numbers m m m m , one can obt ain t he quadric equat ions, w here the unknown variables are [ b1 , b 2, b 3, b ] Som e numerical m et hods m ay be applied f or solving t he equat ions t o realize signals separation B 12( z) A 12( z ) B 21( z ) A 21 ( z ) Experiment results here one can have C11 ( z ) = [ + b12 ( 0) a21 ( 0) ] + [ b12 ( 0) a21 ( 1) + b 12( 1) a2 1( 0) ] e- z + b12 ( 1) a21 ( 1) e - 2z T he algorit hm w as t est ed in t he f ollow ing scenario : T he sources s and s w ere the same speech sig nals as those in Section T he coupling coefficient s corresponding to Eq ( 14) w ere chosen as [ a12( 0) , a12( 1) ] = [ 71, - 53] , C22 ( z ) = [ + b21 ( 0) a12 ( 0) ] + [ b21 ( 0) a12 ( 1) + b 21( 1) a1 2( 0) ] e- z + b21 ( 1) a12 ( 1) e - 2z [ a21( 0) , a21 ( 1) ] = [ 12, 37] For first order filt er channels, b 12( 1) 0, b21 ( 1) 0, a12 ( 1) 0, a21 ( 1) Hence, C11 ( z ) 0, C 22 ( z ) From Eq ( 18) one can know t hat t he theoret ical decoupling coef ficients corresponding t o Eq ( 15) are [ b12 ( 0) , b 12( 1) ] = - [ a12 ( 0) , a12( 1) ] = In conclusion, t o achieve sources separat ion in [ - 71, 53] , the case of f irst order filter channels, it is necessary t o require t hat t he condit ions in Eq ( 21) and [ b21 ( 0) , b 21( 1) ] = - [ a21 ( 0) , a21( 1) ] = Eq ( 24) are satisfied In the case of high order f il- [ - 12, - 37] ters ( great er than one) , one may guess t hat t he By implement ing t he algorit hm in Sect ion 2, one BSS can be realized by using only the correlat ion functions, provided that t he conditions parallel to can obt ain the numerical decoupling coeff icient s as Eq ( 21) and Eq ( 24) are sat isf ied It can be seen [ b 21( 0) , b21 ( 1) ] = [ - 1325, - 3668] 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 T he absolut e errors of t hese coeff icient s bet w een order of filt ers is great er t han one, the terms need to be analy zed further 2 Al gorithm t heoretical and numerical values are g iven as T able T he experim ent result s indicat e t hat num erical decoupling coef ficient s are close to t heoret ical val- T o achieve the decoupling coef ficients b ( 0) , ( 1) , b 21 ( 0) and b21 ( 1) as Eq ( 15) For simb 12 12 [ b12 ( 0) , b 12( 1) ] = [ - 6919, 5019] , ues S ON G Y ou, LIU Zh ong -kan, LI Q i-han 168 Tabl e Errors between theoretical and [ 5] numerical val ues D ecoupling Coef f icient s A bsolut e Errors b12 ( 0) 0181 b12 ( 1) 0281 b21 ( 0) 0125 b21 ( 1) 0032 CJA [ J] Signal Processing , 1994, 36: 287- 314 Y ellin D , W einst ein E Criteria for multich annel signal separat 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 convolutive 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 Processing, 1996, 44: 106- 118 [ 8] on -lin e learning for blind ident if icat ion and blind separation Conclusions of sources [ J ] IEEE T rans on Circuit s an d Syst ems-I, 1996, 43( 11) : 894- 906 T he BSS problem of the tw o channels model is considered in t his paper, w hich is based on cor- [ 9] [ 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 higher -order cum ulant s[ J] Signal Processing , 1999, 73: 267- 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 [ 2] [ J] Signal Processing, 1991, 24( 1) : 1- 10 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 Lindgren U A , Broman H S ource separation using a crit erion based on secon d-order st at ist ics [ J ] IEEE T rans on Signal Processing , 1998, 46( 7) : 1837- 1850 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 Cichochi A , U nbehauen R R obust neural net w orks w it h 276 [ 12] Hyva!rinen A , O ja E Independent component analysis: algorit 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 iterative 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 onautics and A str onautics in 1997 He received his doctoral degr ee in 2003, and then became a teacher there E -mail : song you @ buaa edu cn ... calculation, one can obt ain Crit erion f or Blind S ignals Separat ion Based on Correlat ion Funct ion A ugust 2003 165 mixing sources successf ully BSS for L T I System Separation criterion Consider... signal separation: M et hods an d analysis[ J] IEE E T rans on Signal Processing, 1996, 44: 106- 118 [ 8] on -lin e learning for blind ident if icat ion and blind separation Conclusions of sources... desired signals separation, one can reformulate the problem as that of solving binary quadric nonlinear equat ions L et b = b 12 and b2 = b 21 for simplicit y According t o the noncorrelation bet