The use of the empirical mode decomposition for the identification of mean field aligned reference frames

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1 ANNALS OF GEOPHYSICS, 59, 6, 06, G065; doi:0.440/ag7067 The use of he empiical mode decomposiion fo he idenificaion of mean field aligned efeence fames Mauo Regi *, Alfedo Del Copo, Macello De Laueis Univesià dell Aquila, Dipaimeno di Scienze Fisiche e Chimiche, L Aquila, Ialy Aicle hisoy Received June 6, 06; acceped Ocobe, 06. Subjec classificaion: Mean field aligned efeence fame, Empiical mode decomposiion, Movingaveage, Timeseies analysis. ABSTRACT The magneic field saellie daa ae usually efeed o geocenic coodinae efeence fame. Convesely, he magneohydodynamic waves modes in magneized plasma depend on he ambien magneic field, and is hen useful o oae he magneic field measuemens ino he mean field aligned (MFA) coodinae sysem. This efeence fame is useful o sudy he ula low fequency magneic field vaiaions along he diecion of he mean field and pependiculaly o i. In ode o idenify he mean magneic field he classical moving aveage (MAVG) appoach is usually adoped bu, unde paicula condiions, his pocedue induces undesied feaues, such as specal aleaion in he oaed componens. We discuss hese aspecs pomoing an alenaive and moe efficien mehod fo mean field aligned pojecion, based on he empiical mode decomposiion (EMD).. Inoducion Ulalow fequency (ULF; mhz0 Hz) magneohydodynamic (MHD) plasma waves eceived paicula aenion in he pas yeas [Eisson e al. 006, Pilipeno e al. 008, Agapiov and Cheemnyh 05, Belahovsy e al. 06, Balasis e al. 05]. Geneaed by a vaiey of insabiliies, ULF waves anspo enegy houghou he magneosphee, and may play impoan oles in he enegizaion and loss of adiaion bel paicles (see Men [0] fo a eview). In paicula, ULF waves povide a convenien pobe of he magneosphee, by means of gound [Lichenbege e al. 03] and/o saellies magneic field measuemens [Glassmeie e al. 00, Clausen e al. 009, Regi e al. 03]. Saellie daa ae usually efeed o a geocenic efeence fame, while MHD waves popagaion and popeies can be esablished, using a moe convenien efeence sysem. In his wo we efe o measuemens fom fluxgae magneomees, which povide hee componens of he magneic field, fo boh magneospheic and ineplaneay measuemens. We also assume, wihou losing genealiy, ha he geocenic sola eclipic (GSE) is he oiginal spacecaf efeence fame. Usually, fo saellie mission ha coninuously exploe boh he magneosphee and upseam (foeshoc) egions, he mean field aligned (MFA) coodinae sysem is lagely used fo he oaion pocedue [e.g., Clausen e al. 009, Sais e al. 009, Fancia e al. 0, Fancia e al. 03]. In he sola wind, a echnique widely used o sudy paallel and pependicula magneic field componens of MHD waves is epesened by he minimum vaiance analysis (MVA) [e.g., Tu e al. 989, Klein e al. 99, Buno and Cabone 03]. Howeve, he minimum vaiance diecion does no necessaily coincide wih ha of he ambien magneic field. In paicula, Buno e al. [985] found ha he angle beween he minimum vaiance axis and he mean magneic field diecion anges in he ineval 830 degees, fo heliocenic disances fom 0.9 o 0.88 AU. Ohe mehods compue he aveage magneic field diecion by means of he moving aveage (MAVG) pocedue, ha will be descibed in Secion, and hen apply he MVA on he pojeced componens ino he ohogonal plane wih espec o he mean diecion [Pulinen and Rasäe 009]. In hese cases, he common ey poin concens he compuaion of he main magneic field diecion. ULF waves popey can be associaed wih seveal anisoopy condiions. An impoan issue fo many ypes of anisoopy is he scale ove which he mean magneic field is defined. Timescales of he ambien magneic field and of he peubaion ae cucial quaniies. Fo example, consideing he sola wind measuemens, if ime scales of he flucuaions ae compaable wih he scale of he bacgound flow, ineacions beween he bacgound flow and phase G065

2 REGI ET AL. vaiaions of he flucuaions will influence he bacgound flow [Tu e al. 989]. Fo a local mean field, i migh be he same scale as ha of he consideed flucuaions, o pehaps, 5 o 0 imes ha [Oughon e al. 05]. On his egad, he magneic vaiance anisoopy (see TenBage e al. [0] fo a eview) of he sola wind, as well as he aio beween he ambien magneic field and he magneic field flucuaions [Tu e al. 989, Buno and Cabone 03], can be egaded as useful quaniies o idenify he naue of sola wind ubulen flucuaions. If a ime scale sepaaion exiss wihin he magneic field measuemens, hese ime seies can be hough as a supeposiion of a slowly vaying (ampliude) ambien field B 0 (), an highe fequency signal b() (o peubaion) and an incoheen noise n(): B() B 0 () b() n() () The MFA coodinaes sysem, showed in Figue fo an assigned posiion in he inne magneosphee, is esablished by means of he uni vecos defined as ( indicaes he nom): µ() B 0 ()/ B 0 () z() () B 0 ()/ () B 0 () () o() µ() z() whee µ, z and o ae usually associaed wih compessional, ooidal and poloidal ULF waves modes especively, while () epesens he posiion veco of he spacecaf [Sais e al. 009, Regi e al. 03]. Alhough his definiion is usually used fo magneospheic field measuemens, i may also be exended in upseam egions (see fo example Fancia e al. [0]), bu in his case he z and o componens ae simply elaed wih ansvese oscillaions in he ineplaneay egion (e.g. foeshoc upseam waves). Using he Expessions () we define he insananeous oaion maix fom geocenic o MFA efeence fame as nx() ny() nz() R () zx() zy() zz() (3) pq q u ox() oy() oz() u v ha allows us o pojec he insananeous magneic field veco fom he oiginal geocenic efeence fame ino he MFA one: B MFA () R() B GSE () (4) I is clea fom Equaions () and () ha, in ode o obain he ime seies B MFA () in he MFA efeence Figue. MFA coodinae sysem in he magneosphee: he MFA diecions µ, z and o ae showed ogehe wih he saellie posiion (ed do). The geomagneic field line (blu line) is compued hough T96 magneosphee model [Tsyganeno 995] duing sola quie condiions. sysem, he slowly vaying mean field veco B 0 () mus be found hough an appopiae fileing pocedue. A suiable file should be applied o he daa o emove longe imescale vaiabiliy. Alhough fileing pocedue has he benefi o poenially educe he noise, i should be applied wih cauion, as i may also inoduce aefacs affecing he signals alogehe. In paicula, invese fas Fouie (IFFT) based files should be used only fo saionay and linea phenomena. On his egad, in he MFA oaion pocedue, he moving aveage (MAVG) is a good choice wih espec o any lowpass file, and hence i is used o emove noise and highe fequency flucuaions in a given ime seies (see fo example Clausen e al. [009], Fancia e al. [0], Regi e al. [03]). In applying he MAVG mehod i is assumed ha he chaaceisic flucuaion ime T 0 elaed o B 0 () is much geae han he peiod T b of he peubaion b(). Moeove, T 0 depends on boh saellie moion and naual magneic field vaiaion (e.g. high velociy seam, coonal mass ejecions, cooaing ineacion egions), and could be elaed o non linea and non saionay phenomena. Unde hese condiions he MAVG migh be unsuiable in he oaion pocedue, while a mehod such as he empiical mode decomposiion (EMD), is useful o idenify non linea and non saionay pocesses [Huang e al. 998]. In Secion we discuss on he ininsic poblems involved in he MAVG pocedue on a discee ime seies in he ime domain; in Secion 3 we inoduce a new mehod o idenify B 0 () using he EMD, compaing i o he classical ones (Secion 4); finally, we discuss on he specal modificaion induced by MAVG poce

3 MFA COORDINATES USING EMD due in he MFA coodinae sysem, showing an expeimenal example in Secion 5.. The MAVG effecs The simples pocedue o idenify B 0 () is he esimaion of he aveage ove a ime window T, ceneed a a given ime. By consideing a peiodic peubaion, Equaion () assumes, fo each componen, he following fom B x () B 0,x () b 0x sin(~ x a x ) n x () B y () B 0,y () b 0y sin(~ y a y ) n y () (5) B z () B 0,z () b 0z sin(~ z a z ) n z () whee b 0d (d x, y, z) ae he ampliudes, a d ae he phases, and ~ d /T b,d ae he angula fequencies. Assuming ha T b max{t b,x, T b,y, T b,z } and T 0 min{t 0,x, T 0,y, T 0,z }, and selecing a ime lengh T geae han T b and smalle han T 0 T b T T 0 (6) he aveage of B() aound is (heeafe indicaes he aveage opeao) # T/ B T B() d B T/ since he aveage of he peubaion in a ime ineval T T b is negligible 0 (7) i N B i N! B0i bi ni$ in B b n 0 i i i () ha epesens he pocedue fo he symmeic MAVG compuaion. The main goal of he MAVG is o suppess boh b i and n i ems in Equaion (), so ha B 0 i B i, and Equaion () can be ewien as follows µ i B 0 i / B 0 i z i i B 0 i / i B 0 i () o i µ i z i (hee i epesen he aveaged posiion veco of he spacecaf), obaining he MAVG oaion maix below nx i ny n i z i R i zx i zy z i z i (3) pq q ox i oy o i z i u u v Howeve, he assumpion b i n i 0 due o he MAVG pocedue is no always veified, and each em in Equaion (3) (geneally) depends on he peubaion b i, and lesse on he noise. Figue illusaes qualiaively he MAVG effecs on he peubaion and noise, i.e. b i n i ems in Equaion (). A b T T/ # b() d0 T/ (8) and he aveage noise n T T/ # n() d0 T/ (9) apidly appoaches o zeo, because is auo coelaion ime is sho compaed o ha of he physical phenomena involved [Regi e al. 05, 06]. We now adap he pocedue o discee ime seies, defining he ime i ih, whee h is he sampling peiod and i,,,n, wih n he oal numbe of measuemens. In ode o compue he aveage field aound he i h sample, we use a imewindow T N w h (N )h (0) whee N is he inege numbe of daa on he lef and on he igh wih espec o he i h cenal daa, and hence N w is he window size (i.e. he oal numbe of daa involved in he aveage). Unde his condiion he MAVG, in discee fom, fo he i h em of he magneic field, can be wien as Figue. A movingaveage pocedue example. Top panel shows he oiginal aificial signal composed by nomally disibued whie noise and a 7 mhz (T b 43 s) sine wave. Boom panel shows he esul afe he movingaveage pocedue, fo hee diffeen window sizes (N w ). The sampling peiod is h s. 3

4 REGI ET AL. moe sophisicaed es on he MAVG effecs will be illusaed in Secion 5.. In his es an aificial signal wih h s, obained as he sum of a 7 mhz sine wave of uniay ampliude, ha epesens he peubaion (along an assigned diecion), and a nomally disibued whie noise wih zeo mean and vaiance v 0.5 (op panel), ae consideed. We can see ha he shape of he MAVG (boom panel) songly depends o he choice of he windows size N w, wih ampliudes deceasing wih inceasing N w. In paicula, he noise ampliude deceases moe apidly wih espec o ha of he peubaion, howeve boh quaniies do no compleely vanishes. A quaniaive evaluaion of MAVG effecs on his ind of signals can be found analyically solving Equaion (). Consideing he discee vesion of Equaion (5), and ha he hamonic peubaion a insan i is epesened by he ampliude b i and he angula fequency ~ d /T b,d /N b,d h, Equaion () can be wien as o, a leas, of he ode of log (N s ). If he inequaliy condiion jus above is no saisfied, he sudied ime se i B B gb sins a X d i 0d i 0, d N b, d d (4) In his case, we assumed ha n i ~0, and he movingaveaged b i is compued by means of he elaionship (9) (Appendix A). In Equaion (4) b 0d sin (i/n b,d a d ) is essenially he peubaion b d,i, while g is he imeindependen (dimensionless) faco (see Equaion (9) in Appendix A) esuling in a supeposiion of cosine funcions elaed only wih N w and N b. In geneal, fo assigned N w, g inceases wih N b, while fo assigned N b i deceases wih N w. Moe in deail, local minima (close o bu no equal o 0) ae obained fo N w cn b whee c,, 3,..., c max, and hence when he MAVG window lengh is exacly equal o an inege muliple of N b. Hee, c max is compued as [N s /N b ], whee N s is he oal numbe of samples, and [] is he inege pa opeao. The choice of N w in he MAVG pocedue songly affecs he esimaion of he oaion maix coefficiens. In he saellie efeence fame, he ime flucuaions of he ambien field B 0 () can be due o is ininsic vaiaion, bu also o saellie moion. In he magneosphee, he pevailing ime vaiaions ae ypically due o he saellie moion hough he magneic field lines (excep ansien feaues obseved fo example in he geomagneic ail) while, in he ineplaneay medium, he ambien field vaiaions depend on naual phenomena anging fom coonal mass ejecions (CME) and plasma cloud passages (ha ae moe fequenly obseved duing he highe sola aciviy peiod), as well as cooaing ineacion egions (CIR) lined wih coonal holes (mainly obseved duing he declining phase of he sola cycles). The poblem becomes moe complicaed if we conside he effec of seveal peubaions wih diffeen N b, embedded in he ambien magneic field. In his case, he highe fequency componens (wih lowe N b ) ae emoved fom he ime seies, while he lowe fequency ones (o highe peiodiciy) can affec he pocedue. The MAVG effecs on oaion maix ae discussed in Secion An EMD based pocedue fo lowfequency componens idenificaion An alenaive pocedue o exac he main magneic field fom a signal can be obained using he empiical mode decomposiion (EMD) mehod [Huang e al. 998]. The EMD echnique decomposes a ime seies ino oughly zeomean (muually ohogonal) componens called ininsic mode funcions (IMFs), and a esidual em (Res). Each IMF is found using an appopiae convegen pocedue called sifing algoihm. The sifing pocess uses he uppe e u and lowe e l envelopes compued especively by means of cubic spline inepolaion of he local maxima and minima sepaaely idenified (see Huang e al. [998] fo deail). A he fis sep, he envelopes fom he oiginal daa B() ae compued. By defining e as he mean of he envelopes, he fis exaced componen h is he diffeence beween B() and e [Huang e al. 998], (i.e. B() e h ). This pocedue has o be epeaed seveal imes, unil peassigned sopping cieia ae no saisfied (e.g., Huang e al. [998], Rilling e al. [003], Flandin e al. [004], Wu and Huang [004]). Le assume ha cieia ae saisfied a he h ieaion, he fis ininsic mode funcion (IMF) B is epesened by h. A his sep, he sifing algoihm is applied o B() B, o exac B, and so on. The final mode (heeafe esidue, Res) can sill be diffeen fom zeo; fo daa wih a end, hen he final Res should be ha end, chaaceized a mos by a unique zeo cossing. Afe he EMD decomposiion, he oiginal ime seies can be egaded as supeposiion of all IMFs and he esidue: B() N m Res B whee B (,,...,N m ) ae he IMFs. In paicula, he EMD decomposiion can be egaded as a dyadic file ban [Flandin e al. 004], and hence i is expeced ha he numbe of exaced IMFs (o modes) N m on a ime seies epesened by N s samples, should be N m < log (N s ) (6) (5) 4

5 MFA COORDINATES USING EMD ies possesses a moe complex sucue and a highe infomaion conen han one of a puely sochasic noise [De Michelis e al. 0]. In pinciple, sifing algoihm exacs IMFs wih enegies and imescales (o peiodiciies) ha should be compaed wih ha obained wih pue andomized ime seies (see Wu and Huang [009] and De Michelis e al. [0]). In ode o eain physical meaning in he exacedmodes, diffeen sifing algoihm, wih diffeen sopping cieia ae poposed by seveal auhos (e.g., Huang e al. [998], Rilling e al. [003], De Michelis e al. [0]). In his wo we used he EMD pocedue descibed in Rilling e al. [003] (see also Huang e al. [998]), ha uilizes a sopping cieion fo he sifing algoihm based on wo hesholds i and i. This cieion ensues globally small flucuaions, bu a he same ime, aes ino accoun locally slow lage excusions. Following he auhos we inoduce he mode ampliude a() (e() max e() min )/, whee e() max and e() min ae he envelopes, and he evaluaion funcion v() m()/a(), whee m() is he mean beween hese envelopes. The sifing algoihm is ieaed unil v()< i, fo some assigned facion ( a) of he oal duaion, while v()< i fo he facion a of he oal duaion. Regading he bounday condiions, he algoihm in Rilling e al. [003] minimizes he eo popagaions due o finie obsevaion lenghs by mioing he exema close o he edges. We pefomed in a sepaae analysis, he EMD based oaion algoihm fo he eal case of upseam waves even obseved by Cluse saellie showed in Secion 5., fo seveal sopping paamees i and i, while fo simpliciy we chose a consan oleance of a The esuls (no shown hee) confimed ha ou algoihm does no inoduce significan specal modificaion in he MFA efeence fame assuming i close o 0.05 and i close o 0.5, as suggesed in Rilling e al. [003]. Then, accoding wih he auhos in Rilling e al. [003], we chose as defaul values a 0.05, i 0.05 and i 0i Fo simpliciy, in he following discussion, we will efe o a single componen of he magneic field B i B 0,i b i n i. As explained in he inoducion, we hypohesize ha he B 0 () ime scale is highe han ha of b(), and hence we found useful o define some quaniy in ode o econsuc B 0 () afe he EMD pocedue. Refeing o he peubaion field b(), we seach fo he maximum ime lengh T max unaffeced by ypical peubaion phenomena, defining he equivalen maximum numbe of samples as M max T max /h. Each h IMF can be subdivided in a finie numbe of subinevals, idenified by wo consecuive zeo cossing. Le assume ha each subineval I encloses q I samples. We compue, fo each of hem, he peiod T I q I h, and an enegylined quaniy P I B i I. By means of he pevious quaniies, we evaluae: (a) he aveage peiodiciy of an IMF; T q (7) (b) he aio beween he sum of P I of subinevals chaaceized by a peiodiciy T I > T max (namely P 0 ), and he oal enegy of all subinevals (namely P P I ) R P 0 /P (8) Finally, we econsuc B 0 () EMD simply summing he esidual Res of he EMD mehod and IMFs chaaceized by T I > T max o R > R min, while he emaining IMFs, wih lowe peiodiciies, econsuc [b()n()] EMD. The las condiion ensues ha he IMFs chaaceized by a high enegy even fo a sho ime ineval can be idenified as pa of he main field. The lowe is R min he moe he mehod acceps he IMFs having high mean fequency. We pefomed a Mone Calo es in ode o evaluae he bes value of R min, geneaing 0,000 suogaes of Gaussian whie noise (each one wih v n 0.3), while B 0 and b ae fixed and ae epesened by sinusoids wih peiod geae/lowe han T max. We compued, fo diffeen R min, he coelaion coefficiens beween B 0 and B 0 EMD. The esuls showed ha ou algoihm econsuced B 0 wih > 0.99 in ove 99% of he cases, fo R min 0.. An example is showed in Figue 3, using aificial signals wih h s. Panel (a) shows B 0 () and b()n(), while in he panel (b) he esuling B() B 0 () b() n() is showed. The peubaion is obained windowing a 3 mhz sine wave wih a 000 poins Hanning window ceneed a 000 s. We assume in his simulaion ha B 0 () has a peiod geae han 000 s. Following ou pocedue, he EMD mehod decomposes B() ino 7 IMFs and a Res (panels c); he aveage peiod M of each IMF is also showed. By choosing T max 000 s (i.e. M max 000), idenifying IMF7 and he Res (ed lines) as B 0 (). Using ou mehod, boh B 0 () EMD and [b()n()] EMD ae hen econsuced (panel d). I can be noed ha he oiginal B 0 () is well econsuced, while b()n() can be obained by supeimposing he emaining IMFs. Alhough we used he EMD mehod descibed in Rilling e al. [003], any ind of EMD pocedue can be used following ou mehods, such as he ensemble empiical mode decomposiion (EEMD; Wu and Huang [009]) o he complee ensemble empiical mode decomposiion (CEEMD; Toes e al. [0]). Howeve, we sess ha 5

6 REGI ET AL. Finally we ema also ha, because of he adapive naue of he basis, EMD is ideally suied fo analyzing daa fom nonsaionay and nonlinea pocesses. Howeve, he eade mus beaing in mind ha EMD sill canno esolve he mos complicaed cases, when he pocesses ae nonlinea and he noises also have he same imescale as he signal. In hese cases hei sepaaion becomes impossible (see Wu and Huang [004] fo deails). 4. The compaison beween EMD and MAVG mehods We compaed he EMD and MAVG mehods, basing on he diffeence beween he oiginal signal B 0 () and ha, B 0 h (), obained using diffeen mehods DB 0 ()B 0 () B 0 h () (9) Figue 3. EMD example. (a) Top: synheic ambien field B 0 (), boom: b()n(); (b) he sum of signals in panel a; (c) he IMFs obained using he EMD pocedue, ogehe o he inege numbe of samples ha coesponds o he aveage peiodiciy of each IMF; (d) he exaced B 0 () EMD and [b()n()] EMD. diffeen sifing algoihms wo wih diffeen sopping cieia, and hence, he R min should be evaluaed by means of a dedicaed Mone Calo es fo ohe EMD algoihms. We conclude his secion wih some deails on he selecion cieia examined hee. They ae based on he obseved numeical es esuls, as well as on he diec applicaion on seveal magneospheic and upseam egion evens. Regading T max, i should be chosen accodingly wih wha assumed in Secion. Indeed, if he peiodiciy T b is much smalle han T B, a ime scale sepaaion exiss wihin he magneic field measuemens, and T max will be chosen wihin [T b, T B ]. whee, fo simpliciy, we used he same aificial signal B() showed in Figue 3b. Figue 4 shows he oiginal signal B() (panel a), and DB 0 () compued using he EMD (geen) and he MAVG (blue) pocedues (panel b). In he MAVG mehod a window size of 75 poins is used. As we can see he MAVG mehod is affeced by he pesence of a esidual oscillaion a he same fequency of he oiginal peubaion, bu wih a phase shif of 80, and boh mehod show iegula deviaion fom he expeced signal due o he noise. The esuls ae moe clea by looing a he panel (c) (see also he magnified ime ineval showed in panel d), whee we aveaged he esuls fo 0,000 signals diffeing only fo he whie noise, highlighing hen he mehod dependen effecs. In paicula, he esidual signal (ed) aising fom he MAVG pocedue epoduces he paen of he oiginal peubaion, bu wih a phase shif of 80, while he EMD esidual esuls almos insignifican. The ampliude of he esidual oscillaion and he phase shif depend on he window size used fo he movingaveage. We compued he maximum ampliude diffeence and phase lag beween he oiginal B 0 () and movingaveaged B MAVG () signals, fo seveal values of N w. The maximum diffeence (Figue 5, op panel) geneally deceases wih N w, his end is affeced by oscillaions wih elaive minimum coesponding o inege muliples of he peubaion peiod N b (accodingly wih Figue ). Moeove, he phase lag (Figue 5, boom panel), calculaed via a cossphase analysis a he fequency of he peubaion f b 3 mhz, close o zeo in coespondence of he inevals 0 N b, N b 3N b, ec. The dashed line mas N w 75. Convesely, in a sepaae analysis, we do no found any significan phase lag beween B() and B EMD (). 6

7 MFA COORDINATES USING EMD Figue 4. MAVGEMD compaison. (a) The same aificial signal B() of Figue 3; (b) he diffeence DB 0 () beween he oiginal B 0 () and ha esimaed fom he EMD (geen) and MAVG (blue), whee we used a window size of 75 poins fo he MAVG mehod; (c) he esuling aveage DB 0 () fo 0,000 signals, diffeing only fo a diffeen noise; (d) deail of he phase shif beween oiginal peubaion and he aveaged 0,000 signals [ b()n()] esimaed fom diffeen mehods. Figue 5. MAVG behaviou. Top: ampliude of he oscillaion fo diffeen values of he window size N w. Boom: phase lag beween he expeced signal and he oiginal peubaion, fo he same value of N w. Dashed line coesponds o N w Enegy consevaion es Basing on he pevious esuls i is now clea ha he MAVG affecs he oaion maix coefficiens. In pinciple, he oal enegy should be conseved unde oaion pocedue. Numeically, i should coespond o a vey low diffeence beween powe speca of he oiginal and ha of he oaed signals. In ode o evaluae he enegy diffeence induced in he MFA aligned componens by he MAVG pocedue, we compued a Mone Calo es compaing he powe speca of he ime seies in he oiginal coodinae o ha of he MFA efeence fame. We assumed ha he B x and B z componens of he oiginal signal ae consan values (5 and 6 a.u. especively) wih added ed noise, wih AR().95 (auocoelaion coefficien a lag) and he sandad deviaion v ed 0. especively, ha ae geneaed a each un, while he B y componen is he same signal of Figue 4 excep fo he ed noise geneaed a each un wih popeies jus above descibed. In his es, he saellie posiion is fixed a he coodinaes x, y 0 and z a.u. Figue 6 shows he absolue values of he aveaged powe diffeence DP, compued ove 0 suogaes, as a funcion of N b and N w. The esuls show behavios simila o g (see Figue ) ha affecs he aveaged peubaion b d, and hence DP. Indeed, g apidly inceases fo N w < N b (see Appendix A fo deails), ha coesponds o a violaion of he inequaliy condiion in he lef side of elaion (6). 7

8 REGI ET AL. In ode o evaluae he effecs of he noise, we compued a Mone Calo es using 0,000 diffeen noise eeping fixed he fequency f b 3 mhz and he movingaveage windows size N w 75; he esuling speca ae aveaged o obain he oal PSD aio beween oaed and oiginal signals. The esuls ae shown in Figue 7. EMD mehod gives excellen esuls, showing he same paen of oiginal signal hough he fequencies. Convesely MAVG mehod pesens seveal undesied issues: loweing enegy a low fequencies, ising enegy aound he peubaion fequency, a ponounced pea in coespondence of he fis hamonic of he peubaion and a egula oscillaion a high fequencies. In paicula, he las behavio depend on he window size of he moving aveage, wih a chaaceisic fequency ha incease wih deceasing N w. Figue 6. Absolue values of aveage oal powe diffeence beween oiginal and oaed componens DP as a funcion of samples pe cycle N b and window size N w of moving aveage (see ex fo deails). 5.. The MFA oaion maix insabiliy induced by MAVG pocedue: a numeical example We now discuss how he oaion maix coefficiens depend on he MAVG and EMD. We ema ha, geneally, hese coefficiens depend also on he mean field vaiaion. In ode o simplify ou discussions, we assume a consan ambien B 0 (0, B 0, 0). In addiion, we assume ha he measuing poin (x, 0, 0) is fixed. The chosen condiions will be clea in he nex discussions. Wih hese assumpions and in absence of any peubing signal, each componen of he oaion maix is ime invaian. If we assume b (0, b 0 sin(~, 0) as he peubaion ( using Equaion (4) he oaion maix (3) is given by whee G i R i pq 0 G 0 0 qg i 0 B B g b 0 0 g b 0 0 i 0 Gi 0 u u v sin( ~ sin( ~ (0) We can see ha, geneally, G i swiches fom o, so ha he oaion maix could be unsable inducing undesied feaues. To avoid his poblem, he following condiion mus be saisfied: B b The esuling MFA componens ae 0 0 > g () Figue 7. EMD/MAVG compaison: saisical analysis. 0,000 uns aveage of oal PSD aio beween oaed and oiginal signals by he EMD (op panel) and MAVG (boom panel) mehods. Red lines indicae he uppe and lowe limis of 95% confidence level. Dashed lines indicaes he peubaion fequency and is fis hamonic. B MFA B Gi B0 b0 sin( ~ n! $ B 0 z pq pq qb u u u u q o 0 u v v We can see ha, wih ou assumpion, he MFA 8

9 MFA COORDINATES USING EMD componens ae nonzeo only along he aligned componen B µ G i [B 0 b 0 sin(~]. As discussed in Secion (see also Appendix A fo deails), g deceases wih N w, wih local minima a N w cn b (c,,...,c max ). In he following es we used B nt, b 0 nt, ~ 33 mhz, and h s, so ha N b ~ 30 and B 0 /b The ime seies has a duaion of 3600 s (i.e. i,,, 3600), and is analyzed using FFT algoihm. In he fequency domain we used he elaionships () and (9) in ode o selec wo values of N w equal o and 35: hose values coespond o 0.78 > 0.3 (i.e. unsable condiion) and g Nw g Nw < 0.3 (sable condiion), as i can be seen in he op panel of Figue 8. While he sable oaion maix saisfies he enegy consevaion (i.e. G i cons, lowe panel), he unsable condiion does no peseve enegy and G i is epesened by an asymmeic squae funcion and he negaive values of B y (GSE) become posiive in he esuling B µ, since G i swiches fom o wih ime. Regading he unsable condiion, we show in Figue 9 he oal powe speca of he oiginal and oaed componens, ogehe wih he G i speca. We obseve seveal feaues: (a) enegy aenuaion in he oaed componens (a he signal fequency of 33 mhz and a lowe fequencies); (b) he appeaance of new peas, in Figue 8. Top panel: g as a funcion of N W assuming B nt, b 0 nt, ~ 33mHz, and h s. The unsable (cenal panel) and sable (boom panel) oaion maix em G i and he esuling B µ. Figue 9. Powe specal densiy (PSD) of G i (geen) and B MFA (magena) obained using N w (unsable oaion pocedue, see ex fo deails). The PSD of B GSE is also showed (blac); he fundamenal fequency of he peubaion (33 mhz) and is hamonics ae indicaed by veical doed lines. coespondence o muliples of he fundamenal fequency (i.e. 66 mhz, 99 mhz, ). In a sepaae analysis (no shown hee) we obseved ha he enegy of he even hamonics deceases wih deceasing aio B 0 /b 0, while he enegy peas elaed wih he odd ones incease. In paicula, assuming ha he mean field is zeo (B 0 0), G i becomes a symmeic squae wave funcion, and he fundamenal fequency disappeas, ogehe wih is even hamonics, while only he odd hamonics suvive. Alhough his coesponds o an exeme case sudy, i is ineesing o analyze because i allows us o highligh some of he issues coveed hee. Indeed, in his case he Fouie expansion of B µ b 0 sin(~ is (see Appendix C) B n b 3 0 cos n hi # ~ 4n Q V& n ha clealy conains only he odd hamonics of he fundamenal fequency of he signal. Regading he enegy aenuaion a lowe fequencies, i can be enaively explained by means of he Fouie expansion of B µ G i [B 0 b 0 sin(~], obained using he Fouie expansion of a geneic squae wave G i (see Appendices B and C) wih ampliude A B n 0 0 ~ j~ j~ B c e b c e sin( ) () (3) Fo a pacical pupose, we uncae he Fouie expansion a ode 0, obaining c 0 B 0 c 0 b 0 sin(~) c 0 B y. Since A, fo each coefficien i esuls c <, and hence, indicaing wih P 0 P By () he powe of he 9

10 REGI ET AL. oiginal signal, he esuling oaed componen becomes P B µ P c0by c 0 P 0 P 0. This implies ha only fo c 0 he powe is equal (i.e. P B µ P 0 ), and his is possible only if x and x 0, and hence G() consan (see Appendix B, Equaions (33), (34), (36)), ha coesponds o he validiy of inequaliy condiion (). I is possible o sudy he Fouie seies of B µ following Appendix B fo any ode, bu i is ou of he scope of his wo. The enegy aleaion, due o he MAVG applicaion, can be egaded as a fundamenal elemen of diagnosic in he oaion pocedue, as we will discuss fuhe in he nex secions, by means of an expeimenal example. 5.. A case sudy of he upseam waves even on May 6, 005, obseved by Cluse saellies In ode o compae EMD and MAVG effecs on eal daa, we pesen a case sudy. Basing on he heoeical esuls showed in he pevious secion, we expec ha majo diffeences occu duing ime inevals chaaceized by lage ampliude oscillaions of he geomagneic field (i.e. duing disubed geomagneic condiions). Howeve, i is exemely difficul o find a magneospheic even chaaceized by a lage ampliude oscillaion wih espec o he mean magneic field. Convesely, in he foeshoc egion, i.e. egions upseam he Eah s bow shoc, he ULF upseam waves (000 mhz fequency ange) ampliude is compaable wih he ineplaneay magneic field sengh. In ha egion, when he ineplaneay magneic field maes an angle wih he bow shoc nomal diecion i n,b < 45 (see Regi e al. [04a, 04b]), he exising waves in he sola wind ae amplified by means of he ioncycloon esonance mechanism. We seached fo a ime ineval chaaceized by quasimonochomaic ULF flucuaions in he ansvesal componens wih espec o he ambien ineplaneaymagneic field, inside a daabase of peviously examined evens in Regi e al. [04a]: in ha wo each even was classified by he signalonoise aio, in each GSE componen, using Cluse saellies fluxgae magneomee daa a 4 s sampling peiod [Balogh e al. 00]. We seleced he ime ineval 9 UT on May 6 (DoY 6) of 005, chaaceized by a high signalonoise aio and a low aveage ambien field in he same componen. This condiion should emphasize he effecs induced by MAVG pocedue, heoeically pediced and descibed above. Figue 0 shows he magneic field componens in he GSE efeence fame (op panel), wih oscillaion ampliudes up o ~3 nt, while he aveage magneic field was B 0 ( 4.8, 0.,.) nt. We compued he MFA componens by means of boh EMD and MAVG mehods, also evaluaing he absolue values of he diffeence beween homologous componens DB d B d EMD B d MAVG (whee d µ, z, o), using N w 5 (unsable condiion), as showed in he boom panel of Figue 0. I can be seen ha, duing ime inevals chaaceized by highe oscillaions of B y (i.e. componen wih lowe aveage B 0,y 0. nt), he diffeence becomes moe eviden. Simila, alhough less clea, esuls ae obained using geae N w value in he MAVG pocedue. Figue 0. Upseam wave even obseved by Cluse saellie on May 6, 005. Top panel: oiginal magneic field componens in he GSE efeence fame. Boom panel: he absolue value of he diffeences beween homologous MFA componens obained by means of MAVG and EMD mehods. Duing his ime ineval Cluse was a he posiion (6.5,6.9,7.) Re (Re 6380 m), and aveage magneic field B 0 ( 4.8, 0.,.) nt. In each panel, he zeo levels ae maed by hoizonal doed lines. 0

11 MFA COORDINATES USING EMD In Figue we show a compaison beween oal powe speca (i.e. S GSE ( f ) S x ( f ) S y ( f ) S z ( f )) of oiginal magneic field in he GSE coodinae sysem and ha obained wih EMD and MAVG (i.e. S MFA ( f ) S µ ( f ) S z ( f ) S o ( f )). I can be seen, fo S GSE (maed in boh panels of Figue wih blac cuves), a powe pea a f uw ~5.8 mhz (T b ~ 38.7 s, coesponding appoximaely a N b 9 samples). In his es, we used a numbe of weighs N w 5, 9, 9 fo he MAVG pocedue, while fo he EMD we fixed a cuoff fequency of mhz. Remaable coespondence is found beween S EMD (geen line) and S GSE (blac line). Convesely, S MAVG depas in he lowe fequency ange (i.e. f < f uw ) fo any value of N w, while fo highe fequency and fo N w N b, N b he coespondence is good. Remaable diffeence is found aound f ~ f uw, 3 f uw using N w 5 (i.e. lowe han N b ), accodingly o he heoeical discussions in pevious secion. Fuhe evidences of enegy aleaion as well as phase diffeence in he MAVG pocedue ae fom muliple coheence analysis. Since no exenal noise is inoduced in he oaion pocedue, a muliple coheence c close o is heoeically expeced. We compued c beween oiginal GSE ime seies (3 inpus) and ha in he MFA efeence fame (3 oupus), obained wih boh EMD and MAVG echniques (Figue, boom panel). I can be seen ha c EMD is close o a any fequency, while clea diffeences emege in he c MAVG a seveal fequency bands, fuhe demonsaing ha MAVG ales he powe speca of he oaed componens wih espec o ha of he oiginal ones. 6. Discussions and conclusions In his wo we sudied he effec of he movingaveage pocedue applied o a discee ime seies, compaing he esuls wih hose obained using a new mehod based on he empiical mode decomposiion. Theoeical invesigaions on a simple peiodic and monochomaic peubaion, show ha he MAVG pocedue canno compleely emove he highfequency peubaion fom he mean field, affecing he oaed MFA componens, alhough hey ae imescaled sepaaed. The choice of he window size N w in he MAVG pocedue songly affecs he esimaion of he oaion maix coefficiens. We showed ha he ampliude of he peubaion b(), in he esimaed B 0 (), ae close o bu no equal o zeo only if he window size N w coesponds o a inege muliple of he numbe of sample N b, ha coesponds o he fundamenal peiod of b() (i.e. N w N B,,, ). Moeove, a phase shif of 80 is inoduced by he MAVG pocedue when using a window size N w in he ange beween an odd and Figue. Powe specal densiy S obained by means of EMD algoihm (op panel, geen line) and by means of MAVG algoihm (cenal panel) using diffeen N w (maed wih diffeen colos, cenal panel), ogehe wih he oal powe speca S of GSE componens (blac lines). (boom panel) The muliple coheence c beween GSE and MFA componens using EMD (geen) and MAVG (wih diffeen colos). even inege muliple of he peubaion fundamenal peiod N b (i.e. (m ) N b N w m N b, m,, 3...). These esuls ae also suppoed by means of enegy consevaion es, using a Mone Calo simulaion. We also showed ha heoeically, when he peubaion ampliude is lage wih espec o he mean magneic field, seveal powe peas, a fequencies muliple of he fundamenal one, appea. Howeve i is exemely difficul o found such condiions in he magneosphee, whee he ambien field is highe han ULF waves ampliudes (see fo example Segeev e al. [003]) and he sabiliy condiion is saisfied. Finally, we sudied he MAVG and EMD effecs on eal magneic field daa measued by Cluse saellie on May 6, 005, in he upseam egion. In his case, we found significan diffeences beween he powe speca of he oiginal and ha of oaed componens, using he MAVG.

12 REGI ET AL. In ohe wods, he MAVG opeaes as a lowpass file wih (geneally) a nonzeo phase lag. The choice of appopiae N w is cucial; i should no be oo lage, in ode o be compaable wih he ambien field flucuaion peiod, and no oo small, in ode o ensue ha he powe speca of he esuling oaed MFA componens ae no modified, especially in coespondence o fequency anges of ou inees. Moeove, in some cases, he involved phenomena ae nonlinea and nonsaionay, and hence he MAVG canno be applied o idenify he longpeiodiciies componens of he discee ime seies. A new echnique based on he EMD is pesened; his echnique is applied o aificial (bu almos ealisic) signals, as well as o eal daa: The esuls show ha i allows us o idenify he long peiodiciies componen B 0 wihou phase shif, and enegy modificaions. We fuhe ema ha, in hemavg pocedue, an ideal choice of N w equal o (o muliple of ) N b is no possible because: (a) he peiodiciy of he peubaion T b geneally does no coespond o an inege numbe of samples, i.e. T b /h is no an inege numbe (h is he sampling peiod); (b) eal daa, in he fequency domain, geneally may conain seveal specal peas, each chaaceized by diffeen N b, and hen N w canno be uniquely deemined in ode o suppess, wih he same efficiency, all he oscillaions. Convesely, he EMD pocedue does no inoduce any significan disoion in he MFA componens, pobably due o he adapive naue of he EMD. Howeve, i canno esolve he mos complicaed cases when, fo example, he noise and signals imescale sepaaion does no exis (see Wu and Huang [004] fo deails). The complee codes can be equesed via a he following addesses: mauo.egi@aquila.infn.i; alfedo.delcopo@aquila.infn.i. Acnowledgemens. This eseach aciviy is paially suppoed by he Ialian Pogamma Nazionale di Riceche in Anaide (PNRA PdR03/B09). The auhos acnowledge G. Rilling, P. Flandin and P. Gonçalvès fo shaing hei EMD code (available a hp://peso.enslyon.f/paic.flandin/emd.hml), and A. Ginsed, J. C. Mooe, and S. Jevejeva fo shaing he code o geneae ed noise (available a hp://noc.ac.u/usingscience/cosswavelewavelecoheence/). The auhos also han he Cluse Acive Achive and he FGM expeimen eam. The auhos ae gaeful o R. Buno a Isiuo di Asofisica e Planeologia Spaziali (INAF IAPS) fo helpful discussions, and o he eviewes of his aicle fo hei consucive commens. Refeences Agapiov, O., and O. Cheemnyh (05). Magneospheic ULF waves diven by exenal souces, axiv pepin axiv: Balasis, G., C. Papadimiiou, I.A. Daglis and V. Pilipeno (05). Ulf wave powe feaues in he opside ionosphee evealed by swam obsevaions. Geophysical Reseach Lees, 4 (7), Balogh, A., C.M. Ca, M.H. Acuña, M.W. Dunlop, T.J. Bee, P. Bown, K.H. Fonacon, E. Geogescu, K. H. Glassmeie, J. Hais, G. Musmann, T. Oddy and K. Schwingenschuh (00). The Cluse magneic field invesigaion: oveview of infligh pefomance and iniial esuls, Annales Geophysicae, 9 (0/), 077. Belahovsy, V., V. Pilipeno, D. Mu, E. Fedoov and A. Kozlovsy (06). Modulaion of he ionosphee by Pc5 waves obseved simulaneously by GPS/TEC and EISCAT, Eah, Planes and Space, 68 (), 3. Buno, R., B. Bavassano and U. Villane (985). Evidence fo long peiod Alfvén waves in he inne sola sysem, Jounal of Geophysical Reseach: Space Physics, 90 (A5), Buno, R., and V. Cabone (03). The sola wind as a ubulence laboaoy, Living Reviews in Sola Physics, 0 (), 08. Clausen, L.B.N., T.K. Yeoman, R.C. Fea, R. Behle, E.A. Luce and M.J. Engebeson (009). Fis simulaneous measuemens of waves geneaed a he bow shoc in he sola wind, he magneosphee and on he gound, Annales Geophysicae, 7, De Michelis, P., G. Consolini and R. Tozzi (0). On he muliscale naue of lage geomagneic soms: An empiical mode decomposiion analysis, Nonlinea Pocesses in Geophysics, 9 (6), Eisson, P.T.I., L.G. Blombeg, S. Schaefe and K.H. Glassmeie (006). On he exciaion of ULF waves by sola wind pessue enhancemens, Annales Geophysicae, 4, Flandin, P., G. Rilling and P. Goncalves (004). Empiical mode decomposiion as a file ban, IEEE Signal Pocessing Lees, (), 4. Fancia, P., M. Regi, M. De Laueis, U. Villane and V.A. Pilipeno (0). A case sudy of upseam wave ansmission o he gound a pola and low laiudes, Jounal of Geophysical Reseach (Space Physics), 7, 0. Fancia, P., M. De Laueis and M. Regi (03). Ulf flucuaions obseved along he segma aay duing vey low sola wind densiy condiions, Planeay and Space Science, 8, 748. Glassmeie, K.H., U. Moschmann, M. Dunlop, A. Balogh, M.H. Acuña, C. Ca, G. Musmann, K.H. Fonaçon, K. Schweda and J. Vog, E. Geogescu and S. Buche (00). Cluse as a wave elescope fis esuls fom he fluxgae magneomee, Annales Geophysicae, 9 (0/),

13 MFA COORDINATES USING EMD Huang, N.E., Z. Shen, S.R. Long, M.C. Wu, H.H. Shih, Q. Zheng, N.C. Yen, C.C. Tung and H.H. Liu (998). The empiical mode decomposiion and he Hilbe specum fo nonlinea and nonsaionay ime seies analysis, Royal Sociey of London Poceedings Seies A, 454, 903. Klein, L.W., D.A. Robes and M.L. Goldsein (99). Anisoopy and minimum vaiance diecions of sola wind flucuaions in he oue heliosphee, Jounal of Geophysical Reseach: Space Physics, 96 (A3), Lichenbege, J., M.A. Clilved, B. Heilig, M. Vellane, J. Manninen, C.J. Rodge, A.B. Collie, A.M. Jøgensen, J. Reda, R.H. Holzwoh, R. Fiedel and M. SimonWedlund (03). The plasmasphee duing a space weahe even: fis esuls fom he PLAS MON pojec, Jounal of Space Weahe and Space Climae, 3, A3. Men, F.W. (0). Magneospheic ULF waves: a eview, In: W. Liu and M. Fujimoo (eds.), The dynamic magneosphee, IAGA Special Sopon Boo Seies 3, Spinge, New Yo, 356. Oughon, S., W.H. Mahaeus, M. Wan and K.T. Osman (05). Anisoopy in sola wind plasma ubulence, Philosophical Tansacions of he Royal Sociey of London A: Mahemaical, Physical and Engineeing Sciences, 373 (04), Pilipeno, V.A., O.M. Chugunova and M.J. Engebeson (008). Pc3 4 ULF waves a pola laiudes, Jounal of Amospheic and SolaTeesial Physics, 70 (8), 674. Pulinen, A., and L. Rasäe (009). Minimum vaiance analysisbased popagaion of he sola wind obsevaions: Applicaion o ealime global magneohydodynamic simulaions, Space Weahe, 7 (). Regi, M., P. Fancia, M. De Laueis, K.H. Glassmeie and U. Villane (03). Coheen ansmission of upseam waves o pola laiudes hough magneoail lobes, Jounal of Geophysical Reseach (Space Physics), 8, Regi, M., M. De Laueis and P. Fancia (04a). The occuence of upseam waves in elaion wih he sola wind paamees: A saisical appoach o esimae he size of he foeshoc egion, Planeay and Space Science, 90, Regi, M., M. De Laueis, P. Fancia and U. Villane (04b). The popagaion of ULF waves fom he Eah s foeshoc egion o gound: he case sudy of 5 Febuay 009, Eah, Planes, and Space, 66, 43. Regi, M., M. De Laueis and P. Fancia (05). Pc5 geomagneic flucuaions in esponse o sola wind exciaion and hei elaionship wih elaivisic elecon fluxes in he oue adiaion bel, Eah, Planes and Space, 67 (), 9. Regi, M., M. De Laueis, G. Redaelli and P. Fancia (06). Ulf geomagneic and pola cap poenial signaues in he empeaue and zonal wind eanalysis daa in anacica, Jounal of Geophysical Reseach: Space Physics, (), 8695; 05JA004. Rilling, G., P. Flandin and P. Gonçalves (003). On empiical mode decomposiion and is algoihms, In: IEEEEURASIP Woshop on Nonlinea Signal and Image Pocessing NSIP03. Sais, T., X. Li and H.J. Singe (009). A longduaion naowband Pc5 pulsaion, Jounal of Geophysical Reseach: Space Physics, 4 (A). Segeev, V.A., J.A. Sauvaud, H. Reme, A. Balogh, P. Daly, Q.G. Zong, V. Angelopoulos, M. Ande and A. Vaivads (003). Shap bounday beween he inne magneosphee and acive oue plasma shee, Geophysical Reseach Lees, 30 (5). TenBage, J.M., J.J. Podesa, K.G. Klein and G.G. Howes (0). Inepeing magneic vaiance anisoopy measuemens in he sola wind, The Asophysical Jounal, 753 (), 07. Toes, M.E., M.A. Colominas, G. Schlohaue and P. Flandin (0). A complee ensemble empiical mode decomposiion wih adapive noise, In: Acousics, speech and signal pocessing (ICASSP), 0 IEEE inenaional confeence, Tsyganeno, N.A. (995). Modeling he Eah s magneospheic magneic field confined wihin a ealisic magneopause, Jounal of Geophysical Reseach, 00 (A4), Tu, C.Y., E. Masch and K.M. Thieme (989). Basic popeies of sola wind mhd ubulence nea 0.3 au analyzed by means of elsässe vaiables, Jounal of Geophysical Reseach: Space Physics, 94 (A9), Wu, Z., and N.E. Huang (004). A sudy of he chaaceisics of whie noise using he empiical mode decomposiion mehod, In: Poceedings of he Royal Sociey of London A: Mahemaical, Physical and Engineeing Sciences, 460, Wu, Z., and N.E. Huang (009). Ensemble empiical mode decomposiion: a noiseassised daa analysis mehod, Advances in adapive daa analysis, (0), 4. *Coesponding auho: Mauo Regi, Univesià dell Aquila, Dipaimeno di Scienze Fisiche e Chimiche, L Aquila, Ialy; mauo.egi@aquila.infn.i. 06 by he Isiuo Nazionale di Geofisica e Vulcanologia. All ighs eseved. 3

14 REGI ET AL. Appendices A. Movingaveage applied o a sinusoidal discee ime seies Le s i be a geneic daa seies, he balanced moving aveage s a i h cenal ineval is defined as whee N is he numbe of daa on he lef and on he igh wih espec o he cenal value, used o compue he aveage; hence he oal numbe of daa values used in he aveage is he odd inege N w N. Assume now a peiodic and monochomaic fom fo he seies s s 0 sin(~h a), whee ~ is he pulsaion, a is he phase and h is he sampling peiod. Unde hese assumpions, Equaion (4) a i h ime can be wien as follows In ode o compue he sum (5) we used he igonomeic supeposiion elaionships (poshaphaeesis) fomula (6) obaining he following ems: s s i N i N s in s0 s i sin ~ N hi a! Q V sinq~ hi Q V a V sinq~ hi Q V av sinq~ hi Q V a V sinq~ hi Q V av h sinq~ hi Q NV a V sinq~ hi Q NV av $ h (5) i sin! sin! sin! h sin! p q pq sinp sinq sins X coss X hi Q V $ sin! hi Q V $ sinq~ hi a Vcos! Q~ hv$ hi Q V $ sin! hi Q V $ sinq~ hi a Vcos! Q~ hv$ hi Q 3V $ sin! hi Q 3V $ sinq~ hi a Vcos! 3Q~ hv$ ~ a ~ a ~ a ~ a ~ a ~ a hi Q NV $ sin! hi Q NV $ sinq~ hi a Vcos! NQ~ hv$ ~ a ~ a Theefoe Equaion (5) becomes: s0sinq hi N ~ a V T l cos! l Q~ hv$ Y N (4) (6) (7) Figue. g as a funcion of N w and he pulsaion peiod N b fo a sampled peiod of h (see ex fo deails). The whie dashed lines ma he coesponding minimum g values, associaed wih N w N b, N b, 3N b. In his expession one em is independen and anohe (conaining iindex) is dependen on ime. Moeove, assuming ha he discee ime seies is sampled a h esoluion, we can define he inege numbe of samples N b ha coesponds o he signal peiodiciy T s such ha T s hn b, allowing us o define he angula fequency as ~ /T s /hn b. Afe he subsiuion of his expession in Equaion (7), we obain: s gs sins i 0 Nb b whee we have convenienly defined he ime independen em g Equaion (8) has a imedependen em s 0 sin ( i Nb a) and a imeindependen dimensionless em g. The fome is essenially he oiginal ime seies (o peubaion) s i, and he lae is a supeposiion of cosine funcions depending only on N w and N b. Figue shows he absolue value of g (in a colo scale) as a funcion of N w and N b, fo a sampling peiod h s. g is always diffeen fom zeo excep fo N w N b (,,3,, max ), is indicaed by whie dashed lines. Hee, max is compued as [N s /N b ], whee N s is he oal numbe of samples, and [] is he inege pa opeao. In paicula, g apidly inceases fo N w < N b, exceeding 0.5; his condiion coesponds o a violaion of he inequaliy condiion in he lef side of Equaion (6) (see also Secion 5). i ax w l # cost Y& N w ( N )/ l N b (8) (9) 4

15 MFA COORDINATES USING EMD B. Fouie expansion of a geneic squae wave Assuming an asymmeic squae wave g() as showed in Figue 3 wih a fundamenal peiod T, esuling fom a supeposiion of a posiive pulse, wih ampliude A and duaion x T, and a negaive pulse wih ampliude A and duaion x T (i can be noed ha x x ). Defining ~ /T, he Fouie coefficiens can be calculaed as follows: c ~ # T/ T/ j~ gqve d c c (30) whee c ae he coefficiens fo he posiive pulse, calculaed as c x T/ j ~ x T / while fo he negaive pulse, he coefficiens ae calculaed as and 0,,, 3, Fo Equaion (3) we easy obain (~ /T) c whee we used he elaion jsinx e jx e jx. In ode o calculae he conibuion of he negaive pulse, we use he popeies of he Fouie ansfom of a ime shif in he ime domain F[x( ± x)] X(j~)e ±j~d, sin( x whee X(j~) Ax ) is he Fouie ansfom of x he non shifed, negaive pulse. In ou case he ime shif is d (x x )T/ T/. Finally, fo a ime shifed negaive pulse we obain Using he equaliy e j ( ), and assuming A, he esuling coefficiens fo g(), whee ±, ±, ±3,... ae and hence he Fouie expansion of g() is ~ # Ae d T/ ( x ) T/ j ~ c Ae d ~ ~ # # x T/ j ~ Ae d x T/ ja e / / e j j x x Q V sinqx V Ax x sin c Ax ( x ) x j e sin( x ) c c c x x sin( x ) x Q V x g () 3 c 3 e j~ (3) (3) (33) (34) (35) (36) Figue 3. Asymmeic squae wave. Fom Equaions (33)(36) he Fouie expansion has boh odd (,3,5, ) and even (,4,6, ) fequencies. C. Fouie expansion of he poduc of he sine funcion wih is associaed squae wave Le assume a sinusoidal signal in discee fom s( i ) s 0 sin(~ (a 0) a ime i hi, we define he associaed squae wave q s ( i ) as follows qs() i s() i s() i sin( if G sin( ~ if 0 < hi < n n < hi < n ~ whee n,,3,... In ode o compue he poduc s( i )q s ( i ), we used he Fouie seies expansions of he q s ( i ) odd funcion: 4 qs() i Ssin( ~ 3 sin( 3~ 5 sin( 5~ f n sin( ( n ) ~ X (37) The poduc ŝ() s()q s () assumes he following expession 4s0 s () sin( ~ Ssin( ~ 3 sin( 3~ 5 sin( 5~ 7 sin( 7~ f n sin(( n ) ~ X (38) We compue he poduc of wo sinusoids in Equaion (38) using hewene s fomula sin(x)sin(y) (cos(x y) cos(x y)), obaining he following expessions sin( ~ sin( ~ Q cos( ~ V sin( ~ 3 sin( 3~ 3 Qcos( ~ cos( 4~ V 5 sin( ~ sin( 5~ 5 Qcos( 4~ cos( 6~ V 5

16 REGI ET AL. sin( ~ 7 sin( 7~ 7 Qcos( 6~ cos( 6~ V h sin( ~ n sin( ( n ) ~ n cos(( n ) ~ cos( n~ Q V n sin( ~ sin(( n ) ~ n cos( n~ cos(( n ) ~ Q V (39) Gouping he ems wih he same agumen in Equaion (38) and simplifying he expession we obain he Fouie seies fo he poduc of a sine wave and is associaed squae wave s0 s () " R 3 Wcos( ~ R 5 3 Wcos( 5~ f n R n cos( ~ W f% s0 " 3 cos( ~ 5 cos( 4~ f cos( ~ f% 4n finally obaining s 3 0 sq () s() cos( n. # ~ (40) 4n & n 6