OPTIMAL COMBINATION OF FOURTH ORDER STATISTICS FOR NON-CIRCULAR SOURCE SEPARATION. Christophe De Luigi and Eric Moreau

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1 OPTIMAL COMBINATION OF FOURTH ORDER STATISTICS FOR NON-CIRCULAR SOURCE SEPARATION Chrstophe De Lug and Erc Moreau Unversty of Toulon LSEET UMR CNRS 607 av. G. Pompdou BP56 F-8362 La Valette du Var Cedex France e-mal: ABSTRACT In ths paper we address the problem of blnd source separaton of non crcular dgtal communcaton sgnals. An optmal combnaton of statstcs obtaned from fourth order cumulants that acheves the separaton of non-crcular sources s proposed.. INTRODUCTION In the classcal blnd source separaton problem see e.g. [] [2] [3] and [4] statstcs based matrces or tensors often have an dentcal decomposton. Ths known decomposton s then used through a Jacob-lke algorthm to estmate the so-called mxng matrx. Perhaps one of the most popular algorthms of that knd s gven n []. It s called JADE and ts goal s to jont-dagonalze a set of hermtan matrces. The algorthm n [5] s ntended to jont-dagonalze a set of complex symmetrc matrces. The ICA algorthm n [2] s ntended to dagonalze a fxed order (cumulant) tensor. The STOTD algorthm n [3] s ntended to jontdagonalze a partcular set of (cumulant) thrd order tensor. Actually prncpally n wreless telecommuncaton applcatons non crcular sgnals are of mportance see e.g. [5][6]. Recently the algorthm n [7] has proposed an approach that can combne non-crcular statstcs to crcular one easly for separaton. Notce that the crcular part corresponds to the STOTD algorthm [3] whle the non-crcular one s presented n [7]. In fact wth fourth order complex cumulants whch are often used n source separaton t exsts 3 possbltes of non conjugate statstcs. The man goal of ths paper s to propose an optmal combnaton of these three statstcs through the mnmzaton of the trace of the error matrx of the separaton angles. Frst we compare the optmal combnaton obtaned from our calculus wth the one obtaned through a dscretzaton of the set of the possble combnatons. Second we apply the proposed optmal algorthm and compare t wth each of the three sngle algorthms usng computer smulatons. They llustrate the usefulness to consder an optmal combnaton of the three statstcs. 2. SOURCE SEPARATION In the source separaton problem an observed sgnal vector x[n] s assumed to follow the lnear model x[n] = As[n] () where n Z s the dscrete tme s[n] the (N ) vector of N 2 unobservable complex nput sgnals s [n] {...N} called sources x[n] the (N ) vector of observed sgnals x [n] {...N} and A the (N N) square mxng matrx assumed nvertble. It s classcal to consder that the sources s [n] wth {... N} are zero-mean unt power statonary and statstcally mutually ndependent. We also assume that the sources possess non zero fouth order cumulant (n blnd source separaton t s classcal to consder the four-order cumulant).e. {...N} the 4-th order cumulant Cum{s ( ) [n] s ( )2 [n] s ( )3 [n] s ( )4 [n]} = C ( ) 4 {s } (2) s non zero for all for the consdered optonal complex conjugasons. We also assume that the matrx A s untary. Ths can always be done assumng that a frst whtenng stage s appled onto the observatons. The blnd source separaton problem conssts now n estmatng a untary matrx H n such a way that the vector y[n] = Hx[n] (3) restores one of the dfferent sources on each of ts dfferent components /08/$ IEEE 496

2 Perhaps one of the most useful way to solve the separaton problem conssts n the use of a contrast functons. They correspond to objectve functons whch depend on the outputs of the separatng system and they have to be maxmzed to get a separatng soluton. Let us now propose the followng result. Proposton Usng the notaton then we have J 4c (y) C 4c {a l } 2 = J 4c (a). where J 4c (y) = J 4 (y) wth c complex conjugasons. C ( ) 4 the functon {y j} = Cum{y( ) J 4 (y) = j= s a contrast for whte vectors y. y ( )2 C ( ) 4 {y j} 2 y ( )3 y ( )4 j } (4) Proof: One easly has for all permutaton matrx P J 4 (Py) = J 4 (y) and for all orthonormal dagonal matrx D J 4 (Dy) = J 4 (y). The followng Propostons 2 and 3 fnsh the proof. Proposton 2 For any statstcally ndependent random vector a any orthonormal matrx S we have (5) Proposton 3 For any statstcally ndependent random vectorahavng at most one null cumulant of 4-order J 4 (Sa) = J 4 (a) f and only f S = DP where P s a permutaton and D = dag(d...d N ) such that d 2 =. Proof: The equalty n (6) requres the equaltes N for all j such that C ( ) 4 {a j} 0. Snce j N = s j 4 = = s j 2 = ths s possble f and only f columns j of S have only one non zero component of modulus. That s for at least N columns because we assume that a has at most one null cumulant of 4-order. Because S s orthonormal then S = DP where P s a permutaton and D = dag(d... d N ) such that d 2 =. J 4 (Sa) J 4 (a). (6) Proof: Wth S = (S j ) the mult-lnearty of cumulants and the ndependence of sources we have C ( ) 4 {S j} = N where S ( ) l S ( )2 l S ( )3 l S ( )4 jl C 4c {a l } C 4c {a l } = Cum{a l... a l a l... a l }. }{{}}{{} c terms 4 c terms Now because S s a untary matrx then l l 2 S ml Sml 2 = δ ll 2 m= where δ j = f = j and 0 otherwse. Thus ( N ) J 4c (y) = S l 4 C 4c {a l } 2 and because l = S l 4 = S l 2 = = 3. OPTIMAL COEFFICIENT To mprove the separaton of non-crcular sgnals we propose to use all the avalable statstcs wth the 4-order cumulants. The general statstcal study of ths optmzaton problem seems to be dffcult to acheve. For ths reason we consder a smpler problem : a local asymptotcal analyss n the case of two complex and non-crcular sources. For the fourth order cumulant t exsts 3 non-conjugate possbltes that are: C (0) 4 {y j} = Cum{y y y y j } C () 4 {y j} = Cum{y y y y j } C (2) 4 {y j} = Cum{y y y y j}. For each decomposton we have a dfferent contrast as: J 4 (0)(y) = N J 4 ()(y) = N J 4 (2)(y) = N 4 {y j} 2 C () 4 {y j} 2 C (2) 4 {y j} 2. j= C (0) j= j= (7) (8) 497

3 as The dea here s to combne optmally these 3 contrasts J 4 (opt)(y) = ( λ λ 2 )J 4 (0)(y)+λ J 4 ()(y)+λ 2 J 4 (2)(y) (9) where λ T = (λ λ 2 ) s a real parameter such that λ E λ = {λ [0 ] 2 \ λ + λ 2 [0 ]}. In the case N = 2 these three contrats may be re-wrtten as of e N as N. Assumng θ = ( α φ) to be n the neghborhood of the true parameter θ = ( α φ) we obtan : e(θ)ĥ( θ) = ( θ) where Ĥ( ) and ( ) are respectvely the hessan and the gradent of the emprcal crteron (3) w.r.t θ: C 4 (p)(u) = u T B 4 (p)u (0) wth p = 0 or 2 U s defned by ( ) cos(α) sn(α) exp(jφ) U = () sn(α) exp( jφ) cos(α) ( θ) Ĥ j (θ) = 2 ut θ B λt (0) T () T (2)u = 2 ut θ BλT (0) T () T (2) u θ j +2 u T BλT (0) T () T (2) 2 u θ θ j (5) where α and φ are two angles u s defned by u = (cos(2α) sn(2α)sn(φ) sn(2α)cos(φ)) T (2) and B 4 (p) are real symmetrc matrx gven n [7] for p = 0 n [3] for p = 2 and from ths last one for p =. Thus we drectly have the crteron where C 4 (opt)(u) = u T B λ4 (opt)u (3) B λ4 (opt) = ( λ λ 2 )B 4 (0) +λ B 4 () +λ 2 B 4 (2). (4) wth u θ and 2 u ( = 2 and j = 2) whch can be θ θ j derved. Usng the large number law we show that the emprcal hessan of the optmal crteron converges to ts expectaton that s for the parameter α : 6 Ĉ4 (opt)( ). For the gradent we show that the emprcal one converges to zero and so the central lmt theorem ensures that the emprcal gradent converges to a normal radom varable wth zero mean and a varance whch s on the form : E[ z 2 ]. We have to estmate two angles α and φ accordng to the maxmzaton of C 4 (opt)(α φ) e ( α φ) = argmax C 4 (opt)(α φ). We now focus on the asymptotcal varance property. For ths purpose expectatons n the dfferent matrxb 4 (0) B 4 () B 4 (2) are replaced by sample averages leadng to the emprcal verson of C 4 (opt)( ) whch s denoted Ĉ4 (opt)( ) wth the dfferent expectaton calculated by Ê[y p ] = N y p (k) where N s the number of avalable data. Thus our goal s to fnd an estmate λ of λ through the mnmzaton of the error denoted e n the estmaton of the angles α and φ resultng from the use of the emprcal crteron Ĉ4 (opt)( ). For ths reason we have to determne the lmt In order to allow the calculus of the varance of the emprcal gradent (and to be not too heavy) we choose the value (0 0) for the true parameter θ and we use some approxmatons of the followng knd : s approached by N ỹ p (k) N N z p (k) Ê[ỹ p ] zp (k). We fnd a system of two 2nd degree polynomal equatons n λ and λ 2. All the coeffcents of both equaton depend only on the statstcs of the sources. Fndng the resultant polynomal of 4-degree (t s n fact the ntersecton of two curves) we choose a root whch s real postve and a member of the set [0 ]. The detal of these calculus wll be gven n a forthcomng paper. 498

4 4. COMPUTER SIMULATIONS We llustrate the performances of the proposed algorthm n comparson wth the STOTD algorthm [3] (case where λ = (0 )) and the NC-STOTD one [7] (case where λ = ( 0)) by Monte Carlo smulatons n whch we average over 500 teratons. In our experment we consder two ndependent complex source sgnals whch are non crcular. Frst the objectve s to show and that our asymptotcal varances are good enough estmatons and to show that the optmal combnaton algorthm mproves the two STOTD and the NC-STOTD algorthms. We use 2 dfferent modulatons : a 4 states non-crcular source dstrbuton defned as: Varance of Alpha Performances of the dfferent algorthms & Asymptotc varances Asymp Optmal Asymp NC SOTD Asymp STOTD C3 Asymp STOTD C22 Algo NC STOTD C40 Algo STOTD C3 Algo STOTD C22 Algo Optmal { + j; j; + j; j} wth unform probabltes a 5 states non-crcular source dstrbuton defned as: { + j; j; 0; β ( j); ( j)β} wth the probabltes { 2( + β) ; 2( + β) ; β } β ; 2β( + β) ; 2β( + β) Number of ponts n the sgnals x 0 4 Fg.. Asymptotcal and Emprcal Varance of the dfferent ICA crtera versus the number of samples for a 5 states dstrbuton source obtaned wth the algorthm s better than the asymptotcal one by the fact that our asymptotcal varance s reached through some approxmatons. For each smulaton we gve the emprcal and the asymptotcal varance of the parameter θ obtaned through each crtera. The number of samples goes from 00 to n order to show the effect of N. In Fgure () we have the results for the 5 states dstrbuton. In ths case the performances obtaned by the NC-STOTD crteron are better than the one obtaned by the STOTD crteron. In Fgure (2) we have the results for the 4 states dstrbuton. Ths tme t s the STOTD crteron whch obtaned the better performances n comparson wth the NC-STOTD crteron and the STOTD crteron based on the C () 4 cumulants. In both scenaro we can see that we obtan a better estmaton of θ wth the optmal algorthm. So combnng n an optmal way the dfferent statstc allows to mprove the results of the classcal algorthms n the case of non-crcular sources. Remark We may explaned that sometmes the emprcal varance 5. CONCLUSION Ths paper propose an optmal combnaton of dfferent statstcs from fourth order cumulants n order to separate noncrcular sources. Ths optmal combnaton s reached va an asymptotcal varance calculated on a local Taylor development under some appromatons. It allows to mprove the STOTD and the NC-STOTD algorthm performances. 6. REFERENCES [] J.-F. Cardoso and A. Souloumac Blnd beamformng for non Gaussan sgnals IEE Proceedngs-F vol. 40 pp [2] P. Comon Independent Component Analyss A New Concept? Sgnal Processng vol. 36 pp [3] L. De Lathauwer B. De Moor and J. Vanderwalle Independent Component Analyss and (Smultaneous) Thrd-Order Tensor Dagonalzaton IEEE Transactons on Sgnal Processng vol. 49 pp

5 0 0 Performances of the dfferent algorthms & Asymptotc varances Varance of Alpha Asymp Optmal Asymp NC STOTD C40 Asymp STOTD C3 Asymp STOTD C22 Algo NC STOTD C40 Algo STOTD C3 Algo STOTD C22 Algo Optmal Number of ponts n the sgnals x 0 4 Fg. 2. Asymptotcal and Emprcal Varance of the dfferent ICA crtera versus the number of samples for a 4 states dstrbuton source [4] E. Moreau A Generalzaton of Jont- Dagonalzaton Crtera for Source Separaton IEEE Transactons on Sgnal Processng vol. 49 pp [5] L. De Lathauwer B. De Moor and J. Vanderwalle ICA technques for more sources than sensors Proceedng of the IEEE Sgnal Processng Workshop on Hgher-Order Statstcs (HOS 99) pp June 999 Caesarea Israel. [6] P. Chevaler Optmal array processng for non statonary sgnals n Proc. ICASSP 96 pp Atlanta USA May 996. [7] C. De Lug and E. Moreau Optmal Jont Dagonalzaton of Complex Symmetrc Thrd-Order Tensors. Applcaton to Separaton of Non Crcular Sgnals n Proc. ICA 07 pp London UK September