Extraction of Signals Buried in Noise: Non-Ergodic Processes

Transcription

1 Int J Communications Ntwor and Systm Scincs doi:0436/ijcns0034 Publishd Onlin Dcmbr 00 ( Extraction of Signals Burid in Nois: Non-Ergodic Procsss Abstract Nourédin Yahya By UFR ds Scincs Départmnt d Physiqu Univrsité François Rablais Tours Franc nourdinyahyaby@physuniv-toursfr Rcivd April 5 00; rvisd August 9 00; accptd Octobr 00 In this papr w propos xtraction of signals burid in non-rgodic procsss It is shown that th proposd mthod xtracts signals dfind in a non-rgodic framwor without avraging or smoothing in th dirct tim or frquncy domain Extraction is achivd indpndntly of th natur of nois corrlatd or not with th signal colord or whit Gaussian or not and locations of its spctral xtnt Prformancs of th proposd xtraction mthod and comparativ rsults with othr mthods ar dmonstratd via xprimntal Dopplr vlocimtry masurmnts Kywords: Burid Signals Stationary Non-Ergodic Procsss Spctral Analysis Whit Nois Colord Nois Corrlatd Nois Dopplr Vlocimtry Introduction Quality of signal information in diffrnt aras of scinc is dgradd by ncountrd various naturs of noiss This dgradation may ta diffrnt forms and volvs to obsrvations whr th tim-avragd corrlation function of a procss is diffrnt from th nsmbl-avragd function [] Non-linar filtring and thir corrsponding asymptotic stability ar gnrally proposd to handl such procsss [3-5] In situations whr signals ar totally burid in nois and for which no a-priori information is availabl rgodic hypothsis cannot b validatd Morovr non of th abov mthods is suitabl for xtraction of such burid signals In ral world situations non-rgodic procsss in which a dsird signal is burid may occur as shown by Dopplr vlocimtry masurmnts [6-] usd in this wor It is crucial to notic that by th trms xtraction of signals w man xtraction of clan spctra of burid signals in nois and by burid signals w man signals dfind for low or xtrm low signal-to-nois ratio In [3] w proposd two quivalnt xtraction mthods calld rspctivly modifid frquncy xtnt dnoising (MFED) and constant frquncy xtnt dnoising (CFED) For asy rfrnc lt us rcall that th frquncy xtnt is th intrval 0 f whr f is th sampling frquncy of th burid continuous-tim signal to b xtractd Th first procdur (MFED) is basd on modifying th sampling frquncy and th scond on (CFED) is suitd to a collction of availabl sampl noisy procsss for which th sampling frquncy is constant Clarly CFED offrs implmntation simplicity sinc in most applications signals ar dfind for a constant sampling frquncy Proposd xtraction dos not us any a-priori information on th signal to b xtractd wors without avraging or smoothing in th dirct tim or dual frquncy spac and it is achivd indpndntly of th natur of nois (colord or whit Gaussian or not) and locations of its spctral xtnt Extraction mthods in [3] ar basd on th thory of rgodic stationary procsss Extnsion of ths rsults to xtraction of signals corrlatd with nois in which thy ar burid ar rportd in [4] W hav shown that xtraction of signals corrlatd with nois is achivd without avraging and indpndntly of th natur of nois (corrlatd or not corrlatd or not with th signal colord or whit Gaussian or not) and locations of its spctral xtnt In this wor w xtnd rsults of on of th aformntiond xtraction mthods (CFED) to a mor gnral cas whr rgodic hypothsis givn xprimntal conditions cannot b validatd which is a fact and an issu Copyright 00 SciRs

2 908 N Y BEY In Sction w rcall principal dfinitions and rsults of th CFED xtraction mthod sinc this mthod offrs implmntation simplicity as mntiond abov In Sction 3 xprssion of xtractd spctra of burid signals from non-rgodic procsss is drivd without any sort of avraging or smoothing in th tim or frquncy domain and without assuming th signal uncorrlatd with nois Extraction prformancs and comparativ rsults with othr mthods [5-8] ar discussd and illustratd in Sction 4 It is shown that CFED xtracts th Dopplr man frquncy from xprimntal Dopplr vlocimtry masurmnts for which rgodic hypothsis is not validatd or satisfid Fundamntals [3] In this sction w rcall som principal rsults rportd in [3] Signal Rprsntation z t Lt us considr a finit obsrvation T of z t a procss containing a band-limitd signal burid in zro-man wid sns stationary nois bt in th intrval of lngth T with Tfm ax Th procss z T t availabl at th output of a low-pass filtr of cut-off frquncy f is givn by max 0 bt t st t t T zt t = () 0 othrwis whr bt t and st t rprsnt rspctivly th additiv nois (whit or colord) and th signal obsrvd in th tim intrval of lngth T By considring th instants tn = n f whr f fm ax is th sampling frquncy w can dfin th discrttim procss z ( N n ) with N = Tf Th Sampl Powr Spctral Dnsity (SPSD) [] Givn zn 0 zn zn N stimat f f T T zn n z n f f T N = DFT w can form th whr DFT dnots Discrt Fourir Transform of zn n Hr th stimat dpnds on th frquncy f th sampling frquncy f and th lngth of th obsrvation intrval T It is crucial to notic that () is not a powr spctral dnsity in th usual sns In [] () is dfind as th Sampl powr spctral dnsity or th sampl spctrum of th procss z N n () In th following w rcall for asy rfrnc th CFED xtraction mthod 3 CFED Extraction Mthod Hr w hav a collction of ralizations of duration T of a noisy procss so that th lngth of th total obsrvation intrval is T Ths ralizations dnotd z T t whr p =0 ar now con- p catnatd i ( p z = ) T t zt t pt (3) 3 Sampl Spctrum of Nois W found that th sampl spctrum of nois obtaind by Fourir transformation of (3) is givn by [] p f f T = f p T f T (4) whr f p T f T ar translatd copis of th original sampl spctrum of nois whos componnts ar spacd with th mutual distanc / T on th frquncy axis It is crucial to notic that p in (4) ar rduction factors dfind by p = (5) For th sa of simplicity and without loss of gnrality w lt p =0 = (6) Notic that a justification of (6) is found in [4] Factors p rduc indiffrntly translatd copis th original sampl spctrum of nois indpndntly of thir natur (whit or colord Gaussian or not) and act indiffrntly at all frquncis 3 Spctral Distribution Spctral lins of ach copy of th original sampl spctrum of nois f p T f T of nois ar sparatd by th mutual distanc /T As ths copis ar shiftd by T with rspct to ach othr (s (4)) th rsultd sampl spctrum f f T will xhibit spctral lins sparatd by th mutual distanc T Hnc original spctral lins of nois sparatd by th mutual distanc T ar now distributd in nw frquncy locations cratd in ach frquncy intrval of lngth T On th othr hand th spctrum of th signal st t as givn by th transformation of concatnatd ralizations (3) is spcifid by f f T Sinc zros ar distributd in frquncy locations cratd in ach intrval of lngth T (s []) thn p Copyright 00 SciRs

3 N Y BEY 909 f f T = f f T (7) dpictd by (0) is givn by f f T = f f T f f T S f f T f f T S f f T f f T 33 Extraction Procdur Extraction of th sampl spctrum of th burid signal is obtaind by dcimation This dcimation by th factor is applid in th frquncy domain to th Fourir transformation of (3) Th signal-to-nois ratio of th dcimatd spctrum writtn as a function of th sig- whr f f T S f f T and rprsnt th amplitud spctra of rspctivly th signal and nois nal-to-nois ratio of th original spctrum is givn Hr x is th complx conjugat of x and by f f T is th sampl spctrum of concatnatd ralizations of nois (8) Now lt us find th optimal form undr which xprssion of th CFED sampl spctrum is writtn only as a W hav shown in [] that incrasing incrass th signal-to-nois ratio of th original noisy spctrum f f indpndntly of any corrlation btwn th signal and function of th sampl spctrum of th signal and nois T in which th dsird signal is burid Morovr th varianc of sampl spctral stimats of nois and without avraging in th tim or frquncy nois tnds to zro as incrass domain 3 Non-Ergodic Procsss 3 Sampl Spctrum of Nois Exprssion of th sampl spctrum of nois of a Hr w xtnd abov rsults obtaind for rgodic stationary procsss to procsss for which rgodic hyprssion (4) drivd undr th rgodic assumption In non-rgodic procss is obtaind by using th abov xpothsis cannot b validatd or satisfid W considr (4) on finds translatd copis of th original sampl thrfor that w hav a collction of squncs spctrum of nois dnotd in th rgodic cas by whos probability dnsity functions that dscrib nois f p T f T of ach ralization or sampl affcting thm ar diffrnt from squnc to an othr procss for p =0 on This mans that any sampl procss can b put undr th form (0) it is crucial to notic that w hav ralizations Now in th non-rgodic framwor as dpictd by p p dfind for diffrnt and unnown nois distributions As T t = st t bt t (9) in (0) th indx (p) idntifis ach nois distribution p whr s hr w introduc th indx (p) in ordr to idntify thir T t is th signal dfind abov and b T t is th additiv nois spcifid for th ralization (p) corrsponding original and diffrnt sampl spctra Morovr probability dnsity functions dscribing W can thrfor rwrit (4) by idntifying translatd p b t copis of original sampl spctra of nois by thir corrsponding uppr script p (for p =0 ) as T for p =0 ar assumd unnown p Clarly th procss from which sampls ar T t tan is stationary and not rgodic Covariancs of th follows p sampl procsss T t ar dpndnt on th ( p) sampl procss p (s p 89 of [9] for th dfinition f f T= pp f f T () of non-rgodic procsss) For implmntation simplicity whr w us hraftr th CFED mthod rcalld abov ( p) ( p) whr thos sampls of a non-rgodic procss ar p f f T = f p T f T (3) concatnatd i 3 Dcimatd CFED Sampl Spctrum ( p) ( ) T ()= t st t pt p bt t pt (0) Th dcimatd N-point sampl spctrum applid to f f T as dpictd by () yilds (4) whr is th -dcimation applid to 3 CFED Sampl Spctrum Notic that sinc th powr spctral dnsity of th signal is assumd constant in collctd squncs thn Th sampl spctrum of th procss of duration T as dcimatd sampl spctrum of th signal yilds th sam () f f T = f f T f f T S f f T f f T S f f T f f T (4) Copyright 00 SciRs

4 90 N Y BEY rsult as (7) On th othr hand sinc w hav translatd and diffrnt copis of original sampl spctra of nois thn dcimation in th frquncy domain yilds a spctrum in which contribut cofficints of thos copis of original spctra of nois This mans that th N cofficints of th dcimatd spctrum dscribd by () ar p thos of th translatd copis f p T f T for p =0 Notic that this applis also to th amplitud spctra of nois f f T This mans that by stting p = th dcimatd sampl spctrum of nois in (4) bcoms ( p) f f T = p f p T f T = ; whr th st of th copis 0 is givn by p f p T f T for p =0 (0) ( ) Sinc Fourir cofficints of f f T; 0 ar thos of th translatd copis thn th 0 nois varianc of f f T; is givn by < < (6) min whr min and max ar rspctivly th smallst and th gratst nois variancs containd in th collction of squncs It is crucial to notic that (6) is at th hart of this wor According to (6) w can assum for th sa of simplicity and without loss of gnrality that th dcimatd 0 sampl spctrum of nois f f T; approachs th avrag sampl spctra of th translatd copis 0 of nois Undr this assumption justifid by (6) w can writ th varianc of th dcimatd sampl spctrum of nois f f T; max 0 (0) ( ) f f T as follows p N p c =0 (5) = = (7) whr c for =0 N and p =0 ar Fourir cofficints of th translatd copis 0 Now according to (7) th sampl spctrum 0 f f T; writtn as a function of its Fourir cofficints yilds N (0) ( ) f f T; = c f T (8) 0 =0 whr c = c c rprsnts th avrag of cofficints of translatd copis 0 Notic that th rgodic cas [3] can b obtaind from (8) 0 by stting c = = c By using (7) and (5) th dcimatd sampl spctrum of th procss as givn by (4) is thrfor givn by (0) ( ) = ; (0) ( ) (0) ( ) S f f T f f T S f f T f f T D f f T f f T f f T ; ; whr dcimatd powr spctrum of nois writtn as a function of its amplitud spctra yilds (0) (0) ( ) (0) ( ) f f T; = f f T; f f T; (0) (9) 3 Optimal CFED Sampl Spctrum Lt us in th following find a condition undr which th dcimatd sampl spctrum as dpictd by (9) can b writtn without rctangular trms (cross-products) Th optimal form is that for which th sampl spctrum is dscribd only as function of th spctra of th signal and nois Lt b Fourir cofficints of th amplitud spctrum S f f T of th signal and lt b Fourir cofficints of th amplitud nois spctrum 0 f f ; Sinc f f T = S f f T S f f T () T thn according to (0) and () w hav = = c () whr c is dfind in (8) By using () th sampl spctrum as givn by (9) yilds thrfor xplicitly (0) f f T = f f T; N f T =0 (3) 3 Th Optimal Rduction Factor In th following w driv th optimal rduction factor or th optimal numbr of concatnatd sampl procsss Copyright 00 SciRs

5 N Y BEY 9 undr which contribution of th cross-products in (9) is mad ngligibl Cofficints of th last right-hand sid of (3) can b put undr th form Lt whr yilds = = and not that (4) (5) rwrittn as a function of and By stting c = (6) bcoms c (6) (7) Now lt us find th condition that dfins th minimum valu of undr which (7) is smallr than unity i c (8) W propos to find as a function of th man signal-to-nois ratio of th collction of diffrnt procsss Lt us dfin th man signal-to-nois ratio of th collctd sampl procsss by = ps whr p s rprsnts th man powr of th signal and = 0 is th man varianc of nois of th collctd sampl procsss Hr p is nois varianc of th sampl procss p (for p =0 ) Th man signal-to-nois ratio can b writtn undr th form whr whr = ps I = c I c (9) and c ar arbitrary chosn cofficints and and I = s s=0 s Ic = c / c q q=0 q (30) rprsnt rspctivly th numbr of componnts of th signal and nois It is asy to s that I is boundd by min s I max s (3) whr min s and max s dnot rspctivly th minimum and th maximum valus of th st formd by s for s = 0 and Sinc is an arbitrary chosn cofficint and according to (3) w can considr that I = (3) Similarly Ic = Th signal-to-nois ratio as dpictd by (9) bcoms = (33) c Now th xprssion (8) is satisfid if 4 (34) For a usful intrprtation of (34) w xprss th optimal rduction factor only as a function of th man signal-to-nois ratio By stting =4 min on finds that sinc < two conditions hav to b considrd : 4 and 4 This givs 4 < min (35) 4 min As min in accordanc with (34) thn by choosing xcding th uppr bound of th variation intrval of min or > (36) th condition (8) is fulfilld Hr (36) dpicts th optimal rduction factor as a function of th man signal-to-nois ratio of th collction of non-rgodic sampl procsss This optimal rduction factor rprsnts thrfor th numbr of concatnatd sampl procsss 3 Optimal CFED Spctrum According to (36) (4) yilds (37) By using (37) (3) bcoms > f f T f f T > f This yilds (0) ( ) f T; (38) Copyright 00 SciRs

6 9 N Y BEY max 0 > f f T = f f T = f f T f f T; (39) Hr > rprsnts th numbr of concatnatd sampl procsss undr which th dcimatd sampl spctrum of th procss bcoms optimal or b writtn undr th form (39) It can b sn that undr th condition (36) contribution of th cross-products is mad ngligibl Exprssion of th xtractd spctrum of th burid signal from non-rgodic procsss as dpictd by (39) can b obtaind without th rquirmnt basd on nsmbl avraging and without assuming th signal uncorrlatd with nois Morovr at th limit of larg valus of (39) bcoms lim f f T= f f T (40) max Th sampl spctrum of nois indpndntly of its natur vanishs Th dcimatd spctrum is idntical to th original dtrministic spctrum of th signal Rsults (40) and (39) achiv xtraction of burid spctra of signals from non-rgodic procsss 4 Mthod and Rsults 4 Exprimntal Signals: Dopplr Vlocimtry It is wll nown that Ultrasonic Dopplr vlocimtry provids a non-invasiv mthod for masuring dirction and spd of fluids Information of intrst on Dopplr signals may b found for xampl in [6-] Signals usd hr ar issud from a flow masurmnt apparatus that uss pulsatd mitting sourc with a constant Puls Rptition Frquncy (sampling frquncy) PRF =8 33 Hz A fluid runs at constant or quasi-constant spd and th concrn hr is to masur its man spd [69] Th Dopplr man frquncy f is rlatd to th man vlocity v of th flow by (s [0] among abov rfrncs) f = cos v c (4) whr v is th frquncy of th pulsatd mitting sourc c is th vlocity of th ultrasound wav and dfins th rcivr position with rspct to th dirction of th flow Hr = 0 6 Hz c =500 m/s and = 4 In this xprimnt th man vlocity of th flow is v =4 m/s Th xpctd Dopplr man frquncy is thrfor f =754 Hz 4 Non-Ergodicity of Dopplr Squncs In Figur zro-man four ralizations (a)-(a4) of Dopplr vlocimtry signals carrying information on th spd of th fluid ar dpictd Hr w assum that th natur of nois (whit or colord corrlatd or not) affcting th Dopplr man frquncy is unnown Morovr to accntuat non-rgodicity ffct lt us add to th first Dopplr squnc (a) th third on (a3) and th last on (a4) thr diffrnt colord nois squncs ya n ya3 n and ya4 = 405 a iya nn = un038unn = y n i y n a y n y n y n a3 a3 a3 n spcifid by ya4 n ya4 n ya4 n 09 n 038 n 0 n (4) un is a random signum function (logical func- whr tion which xtracts th sign of a uniformly distributd random numbr) and n is whit Gaussian nois squnc Th lngth of ths squncs is N W assum that signal-to-nois ratios of ths Dopplr squncs ar unnown Howvr man powrs of collctd Dopplr squncs can b computd Ths ar variancs of nois sinc th Dopplr signal carrying th Dopplr man frquncy is vry wa W hav =3 a (4 db) =06 a ( db) =07 a3 ( 55 db) and a 4 =46(5 db) This crats variations of th signal-to-nois ratio According to (39) w hav > Sinc =4 th man signal-to-nois ratio for which an xtraction from nois is possibl is >05( 6 db) If no xtraction is obtaind this mans that th man signal-to-nois ratio of Dopplr squncs is lowr than 6 db In Figur histograms (b) (b) (b3) and (b4) of th four Dopplr squncs ar shown It can b sn that w hav four diffrnt and dformd nois distributions with varying amplituds In (c) (c) (c3) and (c4) covarianc functions of th Dopplr squncs ar dpictd in logarithmic scals for bttr visibility (low corrlation lags) and for comparison purposs Ths covarianc functions ar clarly dpndnt on th sampl Dopplr procss W hav thrfor non-rgodic Dopplr vlocimtry masurmnts (s p 89 of [9]) 43 Extraction Rsults In Figur w propos xtraction of th Dopplr man frquncy by using CFED mthod Wlch PSD stimation [5] and th Thomson's multi-window mthod (MTM) [7] W rcall that MTM uss a ban of windows that comput svral priodograms of th ntir signal and thn avraging th rsulting priodograms to Copyright 00 SciRs

7 N Y BEY 93 Figur Four Dopplr vlocimtry masurmnts (a) (a) (a3) and (a4) in which th Dopplr man frquncy is burid in whit and colord nois Non-rgodic charactr is shown by diffrnt histograms and corrsponding diffrnt covarianc functions (in logarithmic scal) Copyright 00 SciRs

8 94 N Y BEY Figur CFED spctrum and its dcimatd vrsion ar shown in (a) and (b) whr th Dopplr frquncy is xtractd (755 Hz) Comparison with Wlch PSD and th Thomson's multitapr mthod is providd in (c) and (d) construct a spctral stimat In ordr to minimiz th bias and varianc in ach window thss windows ar chosn orthogonal Optimal windows that satisfy ths rquirmnts ar Slpian squncs or discrt prolat sphroidal squncs [8] Now ths four ralizations =4 ar concatnatd and th obtaind procss as givn by (0) is analyzd by th abov mthods Th spctrum of CFED concatnatd ralizations and its dcimatd vrsion =4 ar rspctivly shown in Figurs (a) and (b) Th frquncy of th dpictd spctral lin (Dopplr man frquncy) is f = 755 Hz Th obtaind variation with rspct to th abov xpctd Dopplr frquncy is 03% Th Wlch priodogram stimation shows howvr a widnd pa locatd at 74 Hz with a wa signal-to-nois ratio whras th MTM mthod dpicts with a much war signal-to-nois ratio a spctral lin far from th xpctd Dopplr man frquncy It can b sn that indpndntly of th natur of nois (whit or colord corrlatd or not) affcting xprimntal signals and variation of th signal-to-nois ratio th Dopplr man frquncy is clarly xtractd by CFED from non-rgodic procss with an xcllnt signal-to-nois ratio without any avraging in th tim or Copyright 00 SciRs

9 N Y BEY 95 frquncy domain and without using any a-priori information on th signal (Dopplr man frquncy) and th natur of nois in which th signal is burid 5 Conclusions In this wor xtraction thory of signals burid in non-rgodic procsss is proposd W hav shown that no a-priori information on th signal to b xtractd is usd and no avraging in th dirct tim or frquncy domain is prformd Obsrvd rsults on xprimntal Dopplr vlocimtry masurmnts burid in non-rgodic procsss show that xtraction of th Dopplr man frquncy is achivd indpndntly of th natur of nois corrlatd or not with th signal colord or whit Gaussian or not and locations of its spctral xtnt Obsrvd rsults ar in accordanc with thortical prdictions 6 Acnowldgmnts Th author wishs to xprss his thans to anonymous rviwrs for thir constructiv suggstions 7 Rfrncs [] J L Starc and F Murtagh Astronomical Imag and Data Analysis Springr Vrlag Inc Nw Yor 006 [] P N Pusy and W van Mgn Dynamic Light Scattring by Non-Ergodic Mdia Royal Signals and Radar Establishmnt Physica A Vol 57 No 989 pp [3] A Budhiraja and D Ocon Exponntial Stability in Discrt Tim Filtring for Non-Ergodic Signals Stochastic Procsss and Thir Applications Vol 8 No 999 pp [4] N Oudjan and S Rubnthalr Stability and Uniform Particl Approximation of Nonlinar Filtrs in Cas of Non Ergodic Signals Stochastic Analysis and Applications Vol 3 No pp [5] G Storvi Particl Filtrs for Stat Spac Modls with th Prsnc of Unnown Static Paramtrs IEEE Transactions on Signal Procssing Vol 50 No 00 pp 8-89 [6] W D Barbr J W Ebrhard and S G Kaar A Nw Tim-Domain Tchniqu for Vlocity Masurmnts Using Dopplr Ultrasound IEEE Transactions on Biomdical Enginring Vol BME-3 No pp 3-9 [7] P J Vaitus and R S C Cobbold A Comparativ Study and Assssmnt of Dopplr Ultrasound Spctral Estimation Tchniqus Part I: Estimation Mthods Ultrasound in Mdicin and Biology Vol 4 No pp [8] P J Vaitus R S C Cobbold and K W Johnston A Comparativ Study and Assssmnt of Dopplr Ultrasound Spctral Estimation Tchniqus Part II: Mthod and Rsults Ultrasound in Mdicin and Biology Vol 4 No pp [9] D H Evans W N MacDicn R Sidmor and J P Woodcoc Summary of th Basic Physics of Dopplr Ultrasound Dopplr Ultrasound Physics Instrumntation and Clinical Applications Wily Chichstr 989 [0] J Arndt Estimation of Blood Vlocitis Using Ultrasound Cambridg Univrsity Prss Cambridg 996 [] A Hrmnt and J F Giovanlli An Adaptiv Approach to Computing th Spctrum and Man Frquncy OD Dopplr Signal Ultrasonic Imaging Vol 7 No 995 pp -6 [] N Y By Extraction of Signals Burid in Nois Part I: Fundamntals Signal Procssing Vol 86 No pp [3] N Y By Extraction of Signals Burid in Nois Part II: Exprimntal Rsults Signal Procssing Vol 86 No pp [4] N Y By Extraction of Burid Signals in Nois: Corrlatd Procsss Intrnational Journal of Communications Ntwor and Systm Scincs Vol 3 No 00 pp [5] P D Wlch Th Us of Fast Fourir Transform for th Estimation of Powr Spctra: A Mthod Basd on Tim Avraging ovr Short Modifid Priodograms IEEE Transactions on Audio and Elctroacoustics Vol AU pp [6] S M Kay and S L Marpl Spctrum Analysis A Modrn Prspctiv Procdings of IEEE Vol 69 No 98 pp [7] D J Thomson Spctrum Estimation and Harmonic Analysis Procdings of th IEEE Vol 70 No 9 98 pp [8] Y Xu S Hayin and R J Racin Multipl Window Tim-Frquncy Distribution and Cohrnc of EEG Using Slpian Squncs and Hrmit Functions IEEE Transactions on Biomdical Enginring Vol 46 No pp [9] J S Bndat Masurmnt and Analysis of Random Data John Wily and Sons Inc Hobon 966 Copyright 00 SciRs