Segmentation and tracking of the electro-encephalogram signal using an adaptive recursive bandpass lter

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1 Sgmnttion n trking o th ltro-nphlogrm signl using n ptiv rursiv npss ltr R. R. Ghri 1,2 A. Cihoki 1,3 1 Lortory or Avn Brin Signl Prossing, Brin Sin Institut, RIKEN, Wko-Shi, Sitm, Jpn 2 Assiut Univrsity, Assiut, Egypt 3 Wrsw Univrsity o Thnology, Wrsw, Poln AstrtÐAn ptiv ltring pproh or th sgmnttion n trking o ltro-nphlogrm (EEG) signl wvs is sri. In this pproh, n ptiv rursiv npss ltr is mploy or stimting n trking th ntr rquny ssoit with h EEG wv. Th min vntg inhrnt in th pproh is tht th mploy ptiv ltr hs only on unknown o int to upt. This o int, hving n solut vlu lss thn 1, rprsnts n int istint tur or h EEG spi wv, n its tim untion r ts th non-sttionrity hviour o th EEG signl. Thror th propos pproh is simpl n urt in omprison with xisting multivrit ptiv pprohs. Th pproh is xmin using xtnsiv omputr simultions. It is ppli to omputr-gnrt EEG signls ompos o irnt wvs. Th ptiv ltr o int (i.. th sgmnttion prmtr) is.492 or th lt wv,.36 or th tht wv,.191 or th lph wv,.27 or th sigm wv,.138 or th t wv n.65 or th gmm wv. This implis tht th sgmnttion prmtr inrss with th inrs in th ntr rquny o th EEG wvs, whih provis st on-lin inormtion out th hviour o th EEG signl. Th pproh is lso ppli to rl-worl EEG t or th ttion o slp spinls. KyworsÐEltro-nphlogrm nlysis, Non-sttionrity, Aptiv trking o ntr rquny o iomil signls M. Biol. Eng. Comput., 21, 39, 237±248 1 Introution COMPUTER-AIDED ANALYSIS o th ltro-nphlogrm (EEG) signl, spilly uring slp, is ssntil to ilitt th nlysis o grt numr o ror t (AMIR n GATH, 1989; ANDERSON t l., 1998; NIEDERMEYER n LOPES DA SILVA, 1999; KIM t l., 2; GATH t l., 1992; WRIGHT t l., 199; BLINOWSKA n MALINOWSKI, 1989; ARNOLD t l., 1998; DING t l., 2; ROSIPAL t l., 1998; GOTO t l., 1995; NING n BRONZINO, 1989). Vrious omputr prours mploy prliminry stg o tur xtrtion, ollow y ision-mking or lssi tion n sgmnttion tsks. A lssil prour is to pply th Fourir trnsorm to sussiv winows o th EEG signl to invstigt vrition o th rquny sptrum ovr tim. Th min ssumption ssoit with ths prours is tht th EEG signl ror uring slp or th wking stg Corrsponn shoul rss to Dr R. R. Ghri; -mil: r@sp.rin.rikn.go.jp First riv 22 Sptmr 2 n in nl orm 2 Jnury 21 MBEC onlin numr: # IFMBE: 21 (.g. prorming irnt mntl tsks) is piwis n sttionry. Howvr, owing to th strong non-sttionrity o th EEG signls, th us o ithr sttionrity-s mthos or lok-wis ptiv mthos is otn not stistory or th nlysis o th EEG signl. Vrious tim-vrying utorgrssiv (TVAR) mollings pprohs hv n us or th nlysis n sgmnttionotheegsignl(amirngath,1989;andersontl., 1998; WRIGHT t l., 199; ARNOLD t l., 1998; DING t l., 2; GOTO t l., 1995). In ths pprohs, th EEG signl is ssum to th output o linr ltr y whit-nois pross. Thus th EEG signl is moll s n AR pross o unknown o ints. Ths o ints oul omput ypplyingthlinrpritionrmwork(kaynmarple, 1981; HAYKIN, 1996) to th EEG signl, in ition to n ptiv lgorithm, to hiv on-lin omputtion o th AR o ints. Howvr, suh TVAR molling-s pprohs sur rom tious, hvy omputtion s thy try to upt ll th o ints o th AR mol n to us ths o ints or thon-linomputtionotheegsignlsptrum(anderson tl.,1998;arnoldtl.,1998;dingtl.,2;gototl., 1995). In this ppr, n ltrntiv, int pproh is sri or th sgmnttion n trking o th EEG signl wvs. In this pproh, n ptiv rursiv npss ltr is us to trk th ntr rquny o th EEG signl. Th ptiv npss Mil & Biologil Enginring & Computing 21, Vol

2 ltr mploy hs only on unknown o int to upt. This o int is upt to just th ntr rquny o th ptiv npss ltr to mth with tht o th input signl. Thus th propos wv sgmnttion is s on on-lin stimtion o th ntr rquny o th EEG signl, n th sgmnttion prmtr is sri y th uniqu ptiv ltr o int. Thror th vntg o th prsnt pproh is to mk th sgmnttion n trking pross untion o only on prmtr, proviing simplr n mor urt pproh in omprison with multivrit ptiv pprohs. 2 EEG sgmnttion s on ptiv AR molling To pply th AR rmwork (KAY n MARPLE, 1981; HAYKIN, 1996) to th EEG signl, it is ssum tht th EEG signl is moll s th output o linr ltr y whitnois pross. To omput th AR o ints givn th noisy osrv EEG signl, th linr prition (LP) thory is ppli (HAYKIN,1996).InthLPthory,nstimt^x n othurrnt smpl x n is otin s wight linr omintion o th pst smpls, i.. P ^x n ˆ i x n i iˆ1 Th prition rror o th stimt smpl is givn y n ˆx n ^x n Minimising th sum o squr rrors rsults in th wll-known Yul±Wlkr (YW) qutions, whih shoul solv or th prmtrs i g(kaynmarple,1981;haykin,1996).th rursiv lst squrs lgorithm (RLS) is th tool ommonly us or ning th on-lin solution o th YW qutions. Thus th RLS lgorithm nls us to omput th AR o ints onlin (ptivly). In this s, th ptiv AR o ints i n g n ollow th tim vrying o th EEG signl mol. Unortuntly, th only istint tur tht n ormult using th AR o ints is th powr sptrum o th EEG signl. This on-lin powr sptrum ginst th rquny t tim n n xprss s 1 S jn ˆ 1 P P iˆ1 i n xp j2p i 2 3 Thr r som i ultis ssoit with th AR mollings sgmnttion pproh. Th rst is th unvilility o orr P o th AR mol. Th son is tht th RLS lgorithm onsists o mtrix omputtions, whih inrss th omputtionl ost. Th thir is th hug omputtions rquir or qn 3 t vry instnt o tim n, in ition to thr-imnsionl plottr tht is lso rquir to show th powr sptrum vlus on th rquny±tim gri. 3 Bnpss ptiv ltr 3.1 Filtr strutur Th npss ltr ppli to trk th ntr rquny o npsssignlnthourth-orrbuttrworth ltr,whos trnsr untion is xprss s (RAJA KUMAR n PAL, 1985; 1986; 199) H z ˆ 2 z 2 4 z w n z 1 2 w 2 n 2 z 2 3 w n z 3 4 z whr p ˆ 4 ˆ 1= k 2 2 k 1 2 ˆ 2 1 ˆ 3 ˆ 4 p 1 ˆ 2k 2k 2 2 ˆ 4k 2 2 ˆ 2 k2 1 p 3 ˆ 2k 2k p 2 4 ˆ k 2 2 k 1 k ˆ otn pb w n ˆos p 1 n 2 n os pb 1 (n) ˆ normlis lowr uto rquny s untion o isrt tim n; 2 (n) ˆ normlis highr uto rquny s untion o isrt tim n; n B ˆ normlis nwith o th ltr. From th xprssions givn y qn 5, it is ovious tht, with th ssumption tht th nwith B is onstnt, w n is n only-ntr rquny-pnnt prmtr. Tht is, th npss ptiv ltr H z hs only ntr rqunypnnt o int to upt. It is worthwhil mntioning tht th stility onstrints on H z r provi i 5 k > n jw n j < 1 6 Fig. 1shows th mplitu sptrum jh j2p jginst th rquny or irnt vlus o w n. It is pprnt tht th sptrum movs on th rquny xis to th highr rquny with th rs o w n. This implis tht w n is invrsly proportionl to th ntr rquny o th input signl. Figs 2± show th ltr o int w n ginst th rquny 1 n 2 n or th ltr nwith B qul to.15,.1 n.5, rsptivly. It is ovious tht, or low- n high-rquny rngs, th ltr n unstl. Rltionship jw n j 5 1 xtns with th inrs o th nwith B. Also, it is vint tht this instility ours or normlis rquny rngs rom. to.1 n rom.9 to 1. or B ˆ :1, s n xmpl. Although th rquny rng or whih th instility prolm ours rss with th rs in th nwith B, vry smll nwith is not rommn. To voi this prolm, w propos to shit th rquny o th osrv signl to th mplitu Fig rquny Amplitu sptrum o ptiv rursiv npss ltr t irnt vlus o w(n) (i.. irnt vlus o ntr rqunis): (s) w(n) ˆ.8; () w(n) ˆ.4; ( ) w(n) =.; ( u) w(n) ˆ.4; ( ) w(n) ˆ Mil & Biologil Enginring & Computing 21, Vol. 39

3 w( n ) w( n ) w( n ) Fig ( n)+ 2( n) ( n)+ 2( n) ( n)+ 2( n) Sgmnttion prmtr w(n) ginst twi ntr rquny or irnt ltr nwiths B: () B ˆ.15; () B ˆ.1; () B ˆ.5 stl rquny rng or it is ppli to th ptiv ltr. To show th vntg o th ourth-orr Buttrworth ltr in omprison with othr ltrs, Fig. 3 shows th mplitu sptrum o th xmplry son-orr ptiv ltr with trnsr untion givn y (RAJA KUMAR n PAL, 199). H ALE z ˆ 1 r 2 w n z 1 = r r w n z 1 r 2 z 2 7 whr w n ˆ2r os 2p o n is th only ntr rquny trm in qn 7. Th prmtr r is x sign on rlt to th rquny nwith B, s ollows: B ˆ 1 r =2 Invstigting th mplitu sptrum or nwith B ˆ :1 n or irnt ntr rqunis o shows tht th ltr is not norml npss ltr. It provis unity mplitu t th ntr rquny lon. Thror, y hoosing vry nrrow nwith, this ltr n ppli or trking n nhnmnt o singl sinusoi in whit nois. This pproh hs n rrr to s th ptiv lin nhnr (ALE), owing to th t tht th sptrum o th ltr input signl shows only on lin t th sinusoil rquny, s os th ALE sptrum. 3.2 Aptiv lgorithm Mximising th output powr o th ltr H z mks its ntr rquny sl-just to tht o th input npss signl(rajakumarnpal,1986;199).thtis,thptiv ltr o int w n is upt or th mximistion o th xpt output powr Ey 2 n g. A stnr grintsning pproh oul us or hiving suh mximistion. Th rsulting lgorithm, ll th rursiv mximum mn-squr (RMXMS) lgorithm or upting w n n sri s ollows: From qn 4, th ltr output y n is givn y th ollowing irn qution: y n ˆ x n 2 x n 2 4 x n 4 1 w n y n 1 2 w 2 n 2 y n 2 3 w n y n 3 4 y n Th upt qution inorr or w n to mximis Ey 2 n g is givn y (LJUNG, 1977; RAJA KUMAR n PAL, 1985; 1986). w n 1 ˆw n :5m n H Ey 2 n g 1 1. whr m n > is normlis stp-siz n H Ey 2 n g is th grint with rspt to th ptiv o int w n. It is worthwhil mntioning tht th wll-known rursiv mplitu pr-prossor whit nois ttor i Gn ( )< ε, xn ( ) is whit nois Gn ( ) Fig rquny Amplitu sptrum o son-orr ptiv lin nhnr t irnt ntr rqunis Fig. 4 xn ( ) rquny-shitr npss ptiv iltr y( n) Conptul shm o ptiv pproh or sgmnttion n trking o EEG signl Mil & Biologil Enginring & Computing 21, Vol

4 Fig.5 Gt () mn ( wt ()) wt () mn (G(t)) RsultsopropospprohortrkingwvsgivninTl1in1.BSNRs:()2rlistionsoEEGsignl;()timuntionw(t) o sgmnttion prmtr; () (ÐÐ) mn n (ÐÐ) tru vlus o w(t); () stnr vition o w(t); () tim untion o G(t); () mn o G(t) minimum mn-squr (RMMS) lgorithm is lso givn y qn 1 ut y hnging th sign o th grint. As th mn-squr output Ey 2 n g is unknown, th instntnous grint Hy 2 n g n us s stohsti pproximt or th tru grint. Thus qn 1 n writtn s whr m is x positiv stp-siz n r n is rursiv stimt o th powr o th grint givn y r n ˆlr n 1 2 n 14 w n 1 ˆw n m n y n n 11 with 5l < 1 s th so-ll orgtting tor. whr n ˆH y n. To gurnt th stility o th ltr, w impos n on-lin onstrint w n 1 ˆw n i jw n 1 j 5 1. From qn 9, th grint n n omput s 25 2 n ˆ@y n ˆ 1y n w n y n 2 3 y n 3 1 w n n w 2 n n 2 3 w n n 3 4 n 4 Th normlis stp-siz m n is givn y m nˆm=r n Fig. 6 rquny, Hz Rsults o ptiv AR molling-s pproh or trking wvs givn in Tl 1in 1. B SNR s 24 Mil & Biologil Enginring & Computing 21, Vol. 39

5 Fig.7 mn ( wt ( )) Gt () mn ( Gt ( )) wt () Rsults o propos pprohortrkingwvsgivn in Tl 1in 5. B SNR s: () 2 rlistions o EEG signl;() tim untion w(t) o sgmnttion prmtr; () (ÐÐ) mn n (ÐÐ) tru vlus o w(t); () stnr vition o w(t); () tim untion o G(t); () mn o G(t) 4 Propos pproh n its prtil implmnttion To trk th ntr rquny o th EEG wvs, th EEG signl n pross y th ptiv npss ltr sri in th pring Stion n givn y qn 4. Th ptiv ltr o int w n is thn untion o th ntr rquny 1 n 2 n =2 o h EEG wv, s th nwith o th ltr is hosn to onstnt. This implis tht th tim untion o w n r ts th spontnity o th EEG signl n n us s sgmnttion prmtr. As is vint rom Fig. 2or ltr nwith o.1, w n rss with th inrs o th rquny 1 n 2 n, n it is highly non-linr in th rquny rngs rom. to.3 n rom.7 to 1.. Morovr, it shoul not tht, or th low-rquny rng rom. to.1, w n 51 n, or th high-rquny rng rom.9 to 1., w n 4 1, using th ptiv ltr to unstl or ths rngs. It is lso vint rom Fig. 2tht th rquny proviing th highstility onition n, pproximtly, linr untion or w n is in th rng rom.3 to.7. Unortuntly, th rqunis ssoit with th EEG signl r low rqunis rnging rom.5 Hz to 4 Hz. Thror, with smpling rquny qul to or grtr thn th Nyqust rt, i.. F s 5 8 Hz, th orrsponing normlis rqunis o th EEG signl xist in th instility n non-linr lowrquny rng. To lriy tht, lt th smpling rquny 12.4, or xmpl, n th ltr nwith B.l; thn w n is 1.44 or th lt wv (rom.5 to 3.5 Hz) n.992 or th tht wv (rom 4 to 7 Hz), thus using n instility prolm or th lt wv, n thr is no lr istintion twn oth wvs owing to th smll vrition in th sgmnttion prmtr w n. Thror, to ovrom th instility n non-linr prolms, w n shit th EEG signl rqunis to th rng rom.3 to.7. Anothr ojtiv o this rquny shiting is to nsur tht w n inrss lmost rquny, Hz Fig Rsults o ptiv AR molling-s pproh or trking wvs givn in Tl 1in 5. B SNR s Mil & Biologil Enginring & Computing 21, Vol

6 .65.4 Fig. 9 mn( wt ( )) Gt () mn( Gt ( )) wt () Rsults o propos pproh or trking wvs givn in Tl 2in 1. B SNR s: () 2 rlistions o EEG signl; () tim untion w(t) o sgmnttion prmtr; () (ÐÐ) mn n (ÐÐ) tru vlus o w(t); () stnr vition o w(t); () tim untion o G(t); () mn o G(t) linrly with th inrs in th normlis rquny 1 n 2 n o th EEG signl. A simpl high-rquny shitr is implmnt y moulting th mplitu o singl ton ( rrir signl) using th EEG signl n only pssing th lowr sin moult signl. This n omplish y multiplying th osrv EEG signl y os 2p n n pssing th rsulting signl through npss ltr whos normlis rquny nwith is rom 4=F s to :5=F s. It shoul not tht.5 Hz n 4 Hz, rsptivly, rprsnt th lowst n highst possil rqunis tht n xist in th osrv EEG signl. Thror suh npss ltr is suitl or ll th EEG wvs whn thr is no priori knowlg out th osrv EEG signl. Intuitivly, i w xpt th lowst n highst rquny ssoit with th osrv EEG signl, w n trmin th pproprit nwith, whih in intly improvs th signl-to-nois rtio. I th rrir rquny is,n 1 n 2 n is (twi) th ntr rquny ssoit with th osrv (moulting) EEG signl, thn 2 1 n 2 n will sri (twi) th nw ntr rquny ssoit with th moult signl us s th input o th ptiv npss ltr. Thror, with th inrs o 1 n 2 n, th nw ntr rquny ssoit with th input signl o th ptiv ltr will rs, thus using w n to inrs n vi vrs. Also, th highst nw ntr rquny xists i th rw EEG signl ontins th lowst ntr rquny wv (lt wv), n th lowst nw ntr rquny xists in th gmm wv s. It is vry importnt to trnsr oth th lowst n highst rqunis into th stl rquny rng. It shoul mntion tht hoosing th pproprit smpling n th rrir rqunis nls us to hiv this tsk n thus to lolis w n within th stl n linr rng. W ssum tht itiv nois is whit nois. Thror, in rtin wv n tr onvrgn, i th orthoming wv hs no spi EEG wv, w n nothr lssi r to tt this signl-r nois s. This ttor n hiv s ollows. rquny, Hz Fig Rsults o ptiv AR molling-s pproh or trking wvs givn in Tl 2in 1. B SNR s 242 Mil & Biologil Enginring & Computing 21, Vol. 39

7 .65.4 w(t) Fig. 11 mn (w ( t )) Gt () mn ( Gt ( )) Rsults o propos pproh or trking wvs givn in Tl 2in 5. B SNR s: () 2 rlistions o EEG signl; () tim untion w(t) o sgmnttion prmtr; () (ÐÐ) mn n (ÐÐ) tru vlus o w(t); () stnr vition o w(t); () tim untion o G(t); () mn o G(t) W pss th rw EEG signl through n ptiv Mth-orr linr pritor. Th output rror o th Mth-orr linr prition rror (LPE) ltr is givn y n ˆx n PM g i n x n i iˆ1 15 Th o ints g i n gruptusingthstnrnormlis lst-mn squr (NLMS) lgorithm givn y (HAYKIN, 1996) P M g i n 1 ˆg i n g n x n i x 2 n i iˆ1 16 whr g > is stp-siz. It shoul mntion tht, or onvrgn improvmnt, it is lso possil to us othr mor sophistit linr pritors, suh s th ltti strutur. I th msur EEG signl rprsnts only rnom whit nois, thn th LPE ltr o ints g i n g r los to zro, n o int G n, xprss s G n ˆPM jg i n j < iˆ1 17 is tkn s whitnss ttor. Thn, i G n is lss thn smll positiv thrshol vlu ; x n n onsir to whit nois. In this s, w propos to impos th sgmnttion prmtr to w n ˆA os 2p n. Thror osilltion o th sgmnttion prmtr inits tht th osrv signl is only signl-r nois. This osilltion my provi rquny, Hz Fig Rsults o ptiv AR molling-s pproh or trking wvs givn in Tl 2in 5. B SNR s Mil & Biologil Enginring & Computing 21, Vol

8 .2.5 mn ( wt ( )) Gt () mn ( Gt ()) wt () Fig Rsults o propos pproh or trking ominnt wvs givn in 1. B SNR s: () 2 rlistions o EEG signl; () tim untion w(t) o sgmnttion prmtr; () (ÐÐ) mn n (ÐÐ) tru vlus o w(t); () stnr vition o w(t); () tim untion o G(t); () mn o G(t) inition o th sn o ny o th EEG wvs in th osrv signl. Fig. 4shows onptul shm o th propos ptiv ltring pproh or sgmnttion n trking o th EEG signl wvs. 5 Simultion rsults 5.1 Computr-gnrt t To otin qulittiv vlution o th prsnt ptiv sgmnttion pproh, w trk 2 simult rlistions o irnt EEG signls. Eh EEG signl is ompos o irnt wvs,suhslph,t(stls1n2),t.ehwvis gnrt y pssing zro-mn whit Gussin nois through Hmming-wight FIR ltr o lngth 64, whos nwith is qul to th orrsponing wv nwith. Th powr o h wv is just to unity. W zro-mn whit Gussin nois to hiv 1. n 5. B signl-to-nois rtios (SNRs). Th prmtrs ssoit with th propos ptiv pproh r just s ollows. Th smpling rquny F s n th normlis rrir rquny r 16. n.34 Hz, rsptivly. Ths smpling n rrir rqunis mk th lowst n highst rqunis o 2 1 n 2 n.35 n.655, rsptivly, i.. th vlus o th ptiv ltr Tl 1 Trking irnt EEG wvs: omputr-gnrt EEG signl wvs n orrsponing tru vlus o ptiv o int w(t) Wv Bnwith, Hz Tim rng, s Tru vlu w(t) Dlt.5±3.5 ±5.492 Tht 4.±7. 5±9.36 Alph 7.5±12. 9± Sigm 12.5±15. 12±15.27 Bt 15.5±2. 15±2.138 Tl 2 Trking irnt EEG wvs: omputr-gnrt EEG signl wvs n orrsponing tru vlus o ptiv o int w(t) Wv Bnwith, Hz Tim rng, s Tru vlu w(t) Dlt.5±3.5 ±5.492 WGN ± 5±9 Gmm 2±4 9±13.6 Sigm 12.5±15. 13±16.27 Tht 4.±7. 16± Mil & Biologil Enginring & Computing 21, Vol. 39

9 .2 wt () mn ( wt ()) Fig. 14 Gt () mn( Gt ( )) Rsults o propos pproh or trking ominnt wv in 5. B SNR s: () 2 rlistions o EEG signl; () tim untion w(t) o sgmnttion prmtr; () (±Ð) mn n (- - -) tru vlus o w(t); () stnr vition o w(t); () tim untion o G(t); () mn o G(t) o int w n will within th stl n nrly linr rng. Th normlis nwith o th rquny shitr ltr is thn rom.9 to Th normlis nwith B o th ptiv npss ltr is.1. Th initil vlus o th o ints o th ptiv pproh r just to zro, xpt th rst o int o th whitnss ttor, whih is just to 1.. Th orgtting tor n th stp siz ssoit with th ptiv lgorithm r.95 n.95, rsptivly. Th stp siz hs n slt xprimntlly to hiv st onvrgn n low ututions. Th initil vlu or r is 1.. Rgring th whit-nois ttor, w tk th stp-siz g, th orr M n th thrshol vlu E to.3, 4 n.3, rsptivly. Th mplitu A o th osilltion ssoit with th whitnss ttor is.2. Som rsults o th TVAR molling-s sgmnttion pproh r us or omprisons. Th AR mol (with zro initil o ints) orr P n th orgtting tor o th ptiv RLS lgorithm hv n xprimntlly slt to 8 n.98, rsptivly Trking irnt EEG wvs: To xmin th propos pproh or trking irnt EEG wvs, th pproh is ppli to th EEG signls ompos o th wvs givn in Tls 1 n 2. It shoul not tht th EEG signl givn y Tl 2 ontins signlr whit nois in th tim rng rom 5to 9s. Figs 5, 7, 9 n 11 show th EEG t n th rsults o th prsnt pproh. Th pnls ± o h gur show th 2 rlistions o th EEG signls, th tim untion w t, th mn vlu w t n th stnr vition (SD) o w t, th tim untion o th whitnss ttor G t n th mn vlu G t, rsptivly. Invstigting w t n its mn vlus on rms tht w n intly rognis th irnt wvs o th EEG signl. Also, th tim untion w t n its mn vlus giv us inormtion out th squn o th EEG wvs. Compring th tru vlus o w t givn in Tls 1 n 2 n th vlus o w t shows tht th ptiv pproh provis goo onvrgn proprtis. In th 5. B SNR s, th mn vlu w t shows smll is or lt n tht wvs. This is us th ltr o th rquny shitr hs n just to th wist nwith, whrs oth wvs hv nrrowr nwiths, whih os not improv th SNR so muh. It is lso ovious tht, tr onvrgn, th SD o th sgmnttion prmtr w t is lss thn.5 n rss with th inrs o th SNR. From Figs 9 n 11 it is pprnt tht th whitnss ttor provis int prormn. It is lso ovious tht th sgmn- Mil & Biologil Enginring & Computing 21, Vol

10 -.1 w( t ) mn ( wt ( )) Fig. 15 ( Gt ( )) mn ( Gt ()) Rsults o propos pproh or ttion o omputr-gnrt slp spinls in 1. B SNR s: () 2 rlistions o EEG signl; () tim untion w(t) o sgmnttion prmtr; () (ÐÐ) mn n (- - -) tru vlus o w(t); () stnr vition o w(t); () tim untion o G(t); () mn o G(t) ttion prmtr w t n its mn vlu osillt somwht in th tim whr th r-signl whit nois xists. Figs 6, 8, 1 n 12 show th 256 gry lvls o th vrg powr sptrum, in ils, or th EEG signls otin using th ptiv AR pproh. Although it is ovious tht w n rognis th squn o th EEG wvs, th nwith n th ntr rquny ssoit with h wv r is owing to th itiv nois, spilly in th low SNR suh s 5. B SNR Trking ominnt wv: To show th pility o our pproh in trking th ominnt wv xisting in th EEG signl, w rry out th ollowing simultion. Th osrv signl is ompos o tht n t wvs or prio o 1 s. In th rst 5 s, th powr rtio o th tht wv to th t wv is 1. :.16. In th lst 5s, th powr rtio o th tht wv to th t wv is.16 :1.. Fig. 13 shows th EEG t n th rsults or th 1.B SNR s, whrs Fig. 14 stns or th 5. B SNR s. Invstigting th sgmnttion prmtr w t n its mn vlu shows tht th EEG signl is sgmnt s tht wv in th rst 5 s n s t wv in th lst 5 s. It is lso ovious tht th whitnss ttor shows tht no signl-r nois is osrv Slp-spinl ttion: To xmin th pility o th propos ptiv pproh to tt slp spinls, w rry out th ollowing simultion xmpl. In this xmpl, h rlistion o th omputr-gnrt EEG signls is orgnis s givn in Tl 3. Th wv whos nwith is 6±15Hz orrspons to th slp spinl. Fig. 15 illustrts th EEG t n th rsults or th 1. B SNR s, whrs Fig. 16 shows th 5.B SNR s. Pnls, n o h Figur show th tim untion w t, its mn n its stnr vition, rsptivly. It is ovious tht w n Tl 3 Slp spinl ttion: omputr-gnrt EEG signl wvs n orrsponing tru vlus o ptiv o int w(t) Wv Bnwith, Hz Tim rng, s Tru vlu w(t) Tht 4±7 ±3,7±1,14±17,21±24.36 Slp spinl 6±15 3±7,1±14,17± Mil & Biologil Enginring & Computing 21, Vol. 39

11 -.1 wt () mn ( wt ()) Fig. 16 Gt () mn ( Gt ( )) Rsults o propos pproh or ttion o omputr-gnrt slp spinls in 5. B SNR s: () 2 rlistions o EEG signl; () tim untion w(t) o sgmnttion prmtr; () (±Ð) mn n (- - -) tru vlus o w(t); () stnr vition o w(t); () tim untion o G(t); () mn o G(t) intly rognis slp spinls n nothr wv o th EEG signl y invstigting ithr th sgmnttion prmtr w(t) or its mn. It is pprnt tht, tr onvrgn, th stnr vition is lss thn.3, vn in th 5. B SNR s. Th untion o th mn vlu wå (t) shows tht th ptiv lgorithm ns lss thn 1 s to rh th sty stt. Pnls n o h gur show th whitnss ttion untion G(t) n its mn vlu, rsptivly. It is lr tht oth G(t) n its mn show tht thr is no lvl qul to or lss thn.3, whih on rms tht thr is no signl-r nois xisting in th osrv signl. Th prmtrs o th ptiv ltring pproh r tkn s ollows. Th normlis rrir rquny is.29, th orgtting tor is.9, th stp-siz is.95, n th initil vlus or r() n w() r 1. n, rsptivly. Th normlis rquny nwith o th ptiv npss ltr is tkn to.15. Fig. 17 shows th rst tn hnnls o th msur EEG signls tr ltring y th 1±2 Hz npss ltr. Fig. 17 shows th orrsponing tim untion o th ptiv o int w(t) o ths hnnls. It is vint tht w n tt th slp spinls y invstigting th tim untion o w(t). 5.2 Rl worl t Slp spinl ttion: Aout 12 s roring o 18 hnnls o EEG (Fp1, F8, F4, Fz, F3, F7, T4, C4, Cz, C3, T3, T6, P4, Pz, P3, T5, 2, 1) ws us or monstrting th prormn o th pproh prsnt or th ttion o slp spinls. Eltros wr pl oring to th intrntionl 1±2 systm. Th t wr smpl with smpling rquny o 12.4 Hz. Th msur signls wr ltr y Buttrworth npss ltr twn 1 n 2 Hz. Th signls wr pss orwr n kwr through th ltr to voi phs istortion (ROSAPIL t l. 1998). 6 Conlusion In this ppr, novl ptiv pproh or th sgmnttion n trking o EEG signl wvs hs n prsnt. In this pproh, n ptiv rursiv npss ltr implmnt s ourth-orr Buttrworth ltr is mploy or trking th ntr rquny o h EEG wv. This ltr hs only on unknown o int to upt. This o int, hving n solut vlu lss thn 1, rprsnts n int istint tur or h EEG spi wv, n its tim untion r ts th nonsttionrity hviour o th EEG signl. Thror th min vntg o this pproh is tht Mil & Biologil Enginring & Computing 21, Vol

12 wt () Fig th sgmnttion n trking prosss r hiv using only on prmtr, whih ilitts th nlysis o th EEG signl. Anothr vntg o th pproh is tht th sgmnttion prmtr inrss with th inrs in th ntr rquny o th input signl. This nls us to trk th squn o th EEG signl wvs just y invstigting th hviour o th sgmnttion prmtr. A whit±nois ttor hs lso n introu to invstigt th only nois hypothsis. Th propos pproh hs n ppli to omputr-gnrt t or trking irnt wvs n or slp-spinl ttion. It hs lso n ppli to rlworl EEG t or th ttion o slp spinls n hs n shown to pl o hiving this tsk. AknowlgmntÐTh rvis vrsion o this ppr ws prou in Ntionl Sitm Hospitl (NSH), whr R. R. Ghri sty or out on n hl months. Th uthors woul lik to thnk th st o th ourth oor o th NSH, spilly Dr E. Skizuk, Dr Y. Hoso, Dr T. Tmi n Dr Tkgi, or thir kin hlp n th rinly nvironmnt or to th R. R. Ghri uring his sty. Rrns AMIR, N., n GATH, I. (1989): `Sgmnttion o EEG uring slp using tim-vrying utorgrssiv moling', Biol. Cyrn., 61, pp. 447±455 ANDERSON, C. W., STOLZ, E. A., n SANYOGITA, S. (1998): `Multivrit utorgrssiv mols or lssi tion o spontnous ltronphlogrphi signls uring mntl tsks', IEEE Trns., BME-45, pp. 277±286 ARNOLD, M., MILTNER, W. H. R., WITTE, H., BAUER, R., n BRAUN, C. (1998): `Aptiv AR Moling o nonsttionry tim sris y mns o Klmn ltring', IEEE Trns., BME-45, pp. 553± tim n Rsults o propos pproh or ttion o slp spinls rom rl-worl t: () osrv EEG signls; () tim untion w(t) o sgmnttion prmtr BLINOWSKA, K. J., n MALINOWSKI, M. (1989): `Non-linr n linr orsting o th EEG tim sris', Biol. Cyrn., 66, pp. 159±165 DING, M., BRESSLER, S. L., YANG, W., n LIANG, H. (2): `Shortwinow sptrl nlysis o ortil vnt-rlt potntils y ptiv multivrit utorgrssiv moling: t prossing, mol vlition n vriility ssssmnt', Biol. Cyrn., 83, pp. 35±45 GATH, I., FEUERSTEIN, C., PHAM, D. T., n RONDOUIN, G. (1992): `On th trking o rpi ynmi hngs in sizur EEG', IEEE Trns., BME-39, pp. 952±958 GOTO, S., NAKAMURA, M., n UOSAKI, K. (1995): `On-lin sptrl stimtion o nonsttionry tim sris s on AR mol prmtr stimtion n orr sltion with orgtting tor', IEEE Trns. Signl Pross., 43, pp. 1519±1522 HAYKIN, S. (1996): `Aptiv Filtr Thory' (Prnti Hll, In., Englwoo Clis, N.J.) KAY, M., n MARPLE, S. L. (1981): `Sptrum nlysis ± A morn prsptiv', Pro. IEEE, 69, pp. 138±1418 KIM, H., KIM, T., CHOI, Y., n PARK, S. (2): `Th prition o EEG signls using k-strutur ptiv rtionl untion', Biol. Cyrn., 83, pp. 131±138 KONG, X., BRAMBRINK, A., HANLEY, D. F., n THAKOR, N. V. (1999): `Qunti tion o injury-rlt EEG signl hngs using istns msurs', IEEE Trns., BME-46, pp. 899±91 LJUNG, L. (1977): `Anlysis o rursiv stohsti lgorithms', IEEE Trns. Automt. Contr., AC-22, pp. 551±575 NIEDERMEYER, E., n LOPES DA SILVA, F. (1999): Eltronphlogrphy: si prinipls, linil pplitions n rlt ls, 4th n. (Willims & Wilkins) NING, T., n BRONZINO, J. D. (1989): `Bisptrl nlysis o th rt EEG uring vrious vigiln stts', IEEE Trns., BME-36, pp. 497±499 RAJA KUMAR, R. V., n PAL, R. N. (1985): `A grint lgorithm or ntr-rquny ptiv rursiv npss ltrs', Pro. IEEE, 73, pp. 371±372 RAJA KUMAR, R. V., n PAL, R. N. (1986): `Th rursiv ntrrquny ptiv ltrs or th nhnmnt o npss signls', IEEE Trns. Aoust. Sph Signl Pross., 34, pp. 633±637 RAJA KUMAR, R. V., n PAL, R. N. (199): `Trking o npss signls using ntr-rquny ptiv ltrs', IEEE Trns. Aoust. Sph signl Pross., 38, pp. 171±1721 ROSIPAL, R., DORFFINER, G., n TRENKER, E. (1998): `Cn ICA improv slp spinls ttion?', Nurl Ntw. Worl, 5, pp. 539± 547 WRIGHT, J., KYDD, R. R., n SERGEJEW, A. A. (199): `Autorgrssiv mols o EEG', Biol. Cyrn., 62, pp. 21±21 Authors' iogrphis REDA R. GHARIEB riv his BS, MS n PhD in Eltril Enginring in 1985, 1993 n 1997, rsptivly, rom Assiut Univrsity, Assiut, Egypt. From 1988 to 1997, h ws t th Fulty o Enginring, Assiut Univrsity, working s n Assistnt Lturr. Sin 1997 h hs n lturr. From July 1999 to Jnury 2, h ws Post-Dotorl Fllow with th Fulty o Enginring, Toym Univrsity, Jpn. Sin Jnury 2, h hs n Sintist t th Lortory or Avn Brin Signl Prossing, RIKEN Brin Sin Institut, RIKEN, Jpn. His rsrh intrsts inlu ptiv ltrs, sttistil signl prossing, n highr-orr sttistis. ANDRZEJ CICHOCKI riv his MS (Hons), PhD, n Hilitt Dotort (DrS) in Eltril Enginring rom Wrsw Univrsity o Thnology, Poln, in 1972, 1975, n 1982, rsptivly. Sin 1972, h hs n with th Institut o Thory o Eltril Enginring n Eltril Msurmnts t th Wrsw Univrsity o Thnology, whr h m ull prossor in H is th o-uthor o two ooks n mor thn 15 sinti pprs. H hs spnt svrl yrs s n Alxnr Humolt Rsrh Fllow n Gust Prossor t th Univrsity o Erlngn, Grmny. H is urrntly working t th Brin Sin Institut RIKEN, Jpn, s H o th Lortory or Avn Brin Signl Prossing. Mor tils out his rsrh n oun t: Mil & Biologil Enginring & Computing 21, Vol. 39