A Robust DOA Estimation Based on Sigmoid Transform in Alpha Stable Noise Environment

Transcription

1 MAEC Web of Conferences (08) hps://doi.org/0.05/mecconf/ MIMA 08 A obus DOA Esimion Bsed on igmoid rnsform in Alph ble oise Environmen Li Li * Mingyn e Informion Engineering College Dlin Universiy 66 Dlin Chin Absrc. o suppress lph-sble noise nd co-chnnel inerference his pper defines novel cyclic correlion funcion nd proposes new MUIC lgorihm bsed on sigmoid cyclic correlion funcion. Furhermore he proposed lgorihm (CC-MUIC) is pplied o esime direcion of rrivl (DOA) in lph sble disribuion noise. imulion resuls demonsre h he proposed CC-MUIC cn ge beer performnce hn severl exising lgorihms especilly in he highly impulsive noise environmens. Inroducion ource loclizion by direcion of rrivl (DOA) hs been received considerble enion in rdr sonr nd wireless communicion over he ps yers []. he MUIC lgorihm [] one of he mos well-known subspce mehodology hs rced much enion nd is n sympoiclly unbised DOA esimor bsed on he Gussin noise ssumpion [3 4]. owever in some scenrios noises exhibi n impulsive nure cused minly by sudden burss or shrp spikes [5-7] nd i is inpproprie o model he noise s Gussin disribued. o overcome he Alph sble disribuion noise mny DOA esimion lgorihms bsed on he frcionl [8-0] lower-order sisics (FLO) were proposed. owever he lgorihms hve obined cerin esimion preformion bu hve some limiions: () he chrcerisic exponen of noise mus be esimed o ensure p or 0 p where p is he order of momens nd α is he chrcerisic exponen of he impulsive noise. () hese mehods cn no ccurely esime prmeer if here is no he priori knowledge of chrcerisic exponen furhermore lgorihms preformion degrde seriously even invlid while frcionl lower-order momen vlue is no pproprie. o hndle his problem [] proposes n lgorihm bsed M-esime like correlion lgorihm o esime DOA. Zhng nd Qiu [3] propose he lgorihm of DOA esimion bsed CCO-MUIC. owever performnce of hese lgorihms will degrde significnly when he underlying noise is highly impulsive specilly. Mos mn-mde signls encounered in communicion elemery rdr nd sonr sysems some prmeers do vry periodiclly wih ime. here cn be much o gin in erms of improvemens in performnce of hese processors by recognizing nd exploiing he very specil propery shred by mny mn-mde signls nmed s cyclosionriy [4-6]. he cyclosionry propery cn be exploied o cncel inerference nd bckground noise. here re mny kinds of noise nd inerference in wireless communicion elemery rdr nd sonr sysems nd he received signl my be submerged. o suppress impulse noise nd co-chnnel inerference his pper defines novel cyclic correlion funcion nd propose new lgorihm bsed sigmoid cyclic correlion o esime DOA. he proposed mehod is simple nd effecive. imulions verified he superioriy of he proposed esimion mehod over exising mehods in he presence of boh lph-sble disribuion noise nd cochnnel inerference. ignl model nd sigmoid rnsform.. ignl model Consider L sisiclly independen nrrowbnd plnewve signls impinge on uniformly liner rry (ULA) of M idenicl omni-direcionl sensors. he sensors nd signls re ssumed o be co-plnr since our discussion is confined o zimuh-only sysems. We ssume only K (<L) signls of ineres (OIs) exhibi cyclic propery he cycle frequency. he oher L K signls eiher hve differen cycle frequencies or do no exhibi cyclosionriy. he vecor form of he rry oupu cn be shown s [-] X ( ) = A( ) ( ) + ( ) () where X ( ) = x ( ) x ( ) is he M received M signl vecor ( ) s ( ) s ( ) s ( ) denoes = K rnsmied signl vecor wih he cycle frequency nd A( ) ( ) ( )... ( k ) = conins inerfering sources nd lph sble disribuion noise. Furhermore j sinkd j sin k ( M ) d ( ) = e... e k * Corresponding uhor: dlumsu@gmil.com he Auhors published by EDP ciences. his is n open ccess ricle disribued under he erms of he Creive Commons Aribuion License 4.0 (hp://creivecommons.org/licenses/by/4.0/).

2 MAEC Web of Conferences (08) hps://doi.org/0.05/mecconf/ MIMA 08 ( ) = [ n ( ) nm ( )] is he M noise vecor in which n () m is sequence of i.i.d isoropic complex rndom vrible wih ; cn be convenienly described by is chrcerisic funcion s follows j ( ) =e () where ( 0 is he chrcerisic exponen which cn mesure he heviness of he disribuion il. he smller is he hevier is il is. If = Eq.() could lso describe he Gussin disribuion. he rndom vribles of hve no p-h-order momens when p. is he dispersion prmeer. is he locion prmeer; i depics he men vlue when or he medin vlue when 0.. igmoid funcion he igmoid funcion is widely cceped s common nonliner rnsformion [7]. Is definiion is shown in Eq.(3). ( ) = + exp x ( ) (3) where is he inclined coefficien o djus x ( ) differen scles. hree resuls re obined from he nlysis of he igmoid funcion rnsform [7]: esuls : If xis ( ) process wih = 0 nd = 0 hen ( ) is symmeric disribuion wih zero men in is probbiliy densiy funcion. esuls : If x ( ) is process wih 0 nd = 0 hen we hve w ( ) 0 nd he men vlue of ( ) is zero. esuls 3: If x ( ) is ( ) process wih = 0 hen hs he finie second order momen wih zero men (referred o s second order momen process). A novel cyclic correlion xigmoid ( ) referred o s he igmoid bsed cyclic correlion (CC) is defined in Eq. (4) * jπ xigmoid ( ) = ( + ) ( ) e (4) where represens ime verge. 3 CC-MUIC lgorihm bsed DOA esimion According Eq. (3) nd Eq. (4) we cn obin he sigmoid cyclic correlion funcion ( ) of he received signl vecor X ( ) ( ) = ( + ) ( ) w w e * jπ X X ( ) * ( ) = A A + A igmoid igmoid ( ) ( ) + A + igmoid igmoid (5) where ( ) = ( + ) ( ) igmoid w w e * jπ denoes he sigmoid cyclic uocorrelion of rnsmied signl vecor ( ). We find h he correlion mrix esimion used in subspce lgorihms is replced by he sigmoid cyclic correlion mrix esimion. ince he sigmoid rnsformion cn suppress α-sble noise nd he cyclosionriy of he signl cn selec he OIs he number of sources presened in he sigmoid cyclic correlion mrix is reduced hus he deecion cpbiliy nd resoluion performnce cn be significnly improved. he noise n ( ) is sisiclly independen wih zeromen vlues nd exhibi no cyclosionriy wih cyclic frequency nd he rnsmied signl s ( ) is cyclosionriy independen of noise n ( ) so Eq.(5) cn be rewrien s igmoid ( ) () X = Aigmoid A (6) he singulr vlue decomposiion (VD) of he sigmoid cyclic correlion mrix ( ) ( K M ) ( ) = UV (7) For he rnsmied signl s ( ) is cyclosionriy independen of noise n ( ) rnk ( ( )) = rnk ( igmoid ( )) = K we cn ge lef singulr subspce by he K lef singulr vecors ssocied wih he K nonzero singulr vlues. hen he M K singulr vecors ssocied wih he zero singulr vlues (in prcice here is usully no zero singulr vlue bu insed s smll singulr vlue) spn he noise subspce. he Eq.(6) is denoed by 0 igmoid ( ) X = U U V 0 V (8) = U V U U nd Where V V re uniry nd he digonl elemens of digonl mrix re posiive. he column vecors of U nd U re he eigen vecors spnning he signl subspce nd noise subspce of ( ) respecively wih he ssocied eigen vlues on he digonls of nd. As he signl is independen of he noise nd signl subspces is orhogonl wih noise subspces herefore spil specrum of CC-MUIC cn be go bsed on clssicl MUIC lgorihm which cn be expressed s P( ) = ( ) U U ( ) (9) erching specrl pek of P( ) we cn ge he DOA esimor l. 4 imulion resuls In he simulions we consider ULA wih M = 8 elemens. uppose he crrier frequency of he nrrow-

3 MAEC Web of Conferences (08) hps://doi.org/0.05/mecconf/ MIMA 08 bnd OI wih he DOA ( = 0 nd =40.) is f c = 00 Mz he oher signl is considered s inerference whose crrier frequency is f c = 50 Mz wih DOA = (0 0 ). he percenge bndwidh (he rio of he bndwidh o he crrier frequency) of boh OI nd inerference is ll %. he cyclic frequency of BPK signl is k = f + ( k = 0... ) [5-6] where f0 is he crrier 0 C frequency nd he reciprocl of c is bud re. In his simulion we dop = f c. ince sble disribuion wih deermines infinie vrince we describe he signl-o-noise condiion of using he generlized signl-noise-rio (G). he G is defined by G = 0lg (0) ( s ) where s nd re he vrince of he underlying signl nd dispersion of he noise respecively. he performnce is evlued by wo sisicl performnce mesures: he probbiliy of resoluion nd roo men squre error (ME) of DOA esimion. If he DOA esimion error for ech source is less hn 3 simulneously we cll i successful esimion. he probbiliy of resoluion is he rio of he successful runs o he ol Mone Crlo runs nd he ME of hose successful runs is defined by K K ME = ( l) ( l) L + l= L l= () where nd re he successful esimion of nd nd L is he ol number of successful runs. Ech experimen is execued by 500 Mone-Crlo runs. In his secion some simulions hve been crried ou o compre he performnce of differen mehods including convenionl MUIC [4] FLOM-MUIC [0] CCO-MUIC [5] nd he proposed CC-MUIC. he kernel size is se s = 3 for CCO-MUIC nd he prmeer p is se s p =. for FLOM-MUIC ccording o he simulion resuls nd he conclusions. 4. G In his simulion he generlized signl o noise rio is se s 0 G 0 =.4 nd snpsho number is = 00. he influence of G o he performnce of four lgorihms is shown in Fig.. Alhough ll he performnces re improved wih he increse of G he convenionl MUIC mehod is inferior o oher four mehods. For low Gs boh he CCO-MUIC nd he CC-MUIC cn ge higher successful probbiliy hn oher mehods however he CC-MUIC cn obin more ccure DOA esimion resuls. For high Gs he CCO-MUIC nd MCC-MUIC cn obin smller MEs hn oher mehods. herefore he CC-MUIC lgorihm hve beer preformion especilly when he G vlues re low. () Probbiliy of resoluion (b) oo men squre error Fig.. Probbiliy of resoluion nd ME versus G () Probbiliy of resoluion (b) oo men squre error Fig.. Probbiliy of resoluion nd ME versus npsho 3

4 MAEC Web of Conferences (08) hps://doi.org/0.05/mecconf/ MIMA npsho In his simulion he generlized signl o noise rio is se s G = 0dB nd =.5. he influence of he npsho o he performnce of four lgorihms re shown in Fig.. From Fig. we cn find he ME of oher hree mehods decrese wih he increse of snpsho number excep condiionl MUIC mehod. o we dp = 800 for furher simulions. Furhermore hese hree mehods hve similr probbiliy of resoluion. owever he proposed mehod cn obin lower esimion ME. 4.3 Chrcerisic exponen In his simulion he generlized signl o noise rio is se s G = 0dB nd 0.. he influence of he Chrcerisic exponen o he performnce of four lgorihms re shown in Fig.3. We cn lso see h he performnce of he proposed mehod vries slighly wih he vriion of α nd he proposed mehod cn ge significnly higher successful probbiliy nd beer esimion of DOA hn hose of oher mehods when. I mens h he proposed mehod is more robus o highly impulsive noise. We lso cn find h he esimion ccurcy of he proposed mehod is higher hn oher mehods () Probbiliy of resoluion (b) oo men squre error Fig.3. Probbiliy of resoluion nd ME versus 5 Conclusion Inspired by he heory of igmoid funcion his pper defines he sigmoid cyclic-correlion funcion (CC) nd consrucs cyclic correlion mrix bsed on he sigmoid rnsformion. he corresponding new esimion mehod (CC-MUIC) is proposed for DOA esimion under he -sble noise condiions. We compre performnce of he proposed mehods wih hose of condiionl MUIC FLOM-MUIC nd CCO-MUIC lgorihms under he condiion of complex isoropic symmeric sble noise model. he simulion resuls show h he proposed mehod cn ge beer esimion resuls. Acknowledgemens his work ws prly suppored by he ionl url cience Foundion of Chin under Grns nd he Ph.D Progrms Foundion of Lioning Province of Chin nd he Chin cholrship Fund Progrm. eferences.. Krim M.Viberg IEEE ignl Proc. Mg. 3 4 (996)... O. chmide. IEEE. Anenn. Propg. 34 3(986). 3. B. Por B. Friedlnder IEEE rns. Acous. peech ignl Processing (988). 4. P. oic. Arye IEEE rns. Acous. peech ignl Process. 375 (989). 5. M.D. Buon J.G. Grdiner I.A. Glover IEEE rns. Veh. echnol. 5 3 (00). 6. Xu Z. Yng C. n Z. nd heng Z. IEEE Commun. Le. 0 (07). 7.. Ghofrni IE ignl Process. 8 5 (04). 8. M. ho C.L. ikis Proc. IEEE. 8 7 (993). 9. P. sklides C.L. ikis IEEE rns. ignl Processing.447 (996). 0. L. sung-sien J.M. Mendel IEEE rns. ignl Process (00).. J.F. Zhng.. Qiu. Li Ac Elecronic inic 43 3 (05).. J.F. Zhng.. Qiu A.M. ong. ng ignl Process. 04 (04) Qiu J. F. Zhng A. M. ong ignls Process. 8 4 (0). 4..V. chell.a. Clbre W.A. Grdner B. G. Agee IEEE rns. Ac. p. ignl Process (989). 5. W.A. Grdner IEEE rns. Ac. p. ignl Process (986). 6. W.A. Grdner W.A. Brown C.K. Chen IEEE rns. Commun (987). 4

5 MAEC Web of Conferences (08) hps://doi.org/0.05/mecconf/ MIMA Qiu. Wng Y. Zhng. Mehods Archive 44 (00). 5