Transient Multiexponential Data Analysis Using A Combination of ARMA and ECD Methods

Transcription

1 4 th Intrnatonal Confrnc on Computr Engnrng and Tchnology (ICCET IPCIT vol.4 ( ( IACIT Prss, ngapor Transnt ultxponntal Data Analyss Usng A Combnaton of ARA and ECD thods Abdussamad U. Jba, omoh-jmoh E. alam Dpt. of Elctrcal Engnrng, Bayro Unvrsty Kano, Ngra Dpartmnt of chatroncs Engnrng, Intrnatonal Islamc Unvrsty alaysa Abstract. Anothr attmpt at stmatng th tm constants and numbr of componnts of multpl xponntals n wht Gaussan nos s prsntd. Basd on classcal Gardnr transform, th approach conssts of two tchnqus. Frst, xponntal compnsaton dconvoluton mthod s usd to dconvolvd th dscrt convoluton modl arsng from th applcaton of Gardnr transform. Th dconvolvd data s thn truncatd and furthr procssd usng autorgrssv movng avrag (ARA modl whos AR paramtrs ar dtrmnd by usng hgh-ordr Yul-Walkr quatons va th sngular valu dcomposton (VD algorthm. mulatons carrd out usng a numbr of synthtc sgnals dmonstrat th ffctvnss of th proposd tchnqu. mulaton rsults shows that ths combnaton s mor ffctv than many xstng tchnqus. It s clarly dmonstratd that th proposd approach suprsds a numbr of popular tchnqus. Its lmtatons ar also hghlghtd. Kywords: ARA, ECD, dconvoluton, Gardnr transform, VD. Introducton asurmnt and analyss of transnt multxponntal data s an actvty that has attractd th attnton of many rsarch workrs n appld scnc and ngnrng. ultxponntal transnt data ar ncountrd n smconductor physcs (dp-lvl transnt spctroscopy, bophyscs (fluorscnc dcay analyss, nuclar physcs and chmstry (radoactv dcays, nuclar magntc rsonanc, tc. In ths and smlar applcatons, th data sgnal s of th form λ ( = A + n( = ( whr n ( s an addtv wht nos and A, λ, and ar unknown paramtrs to b dtrmnd. Unlk th usual spctrum analyss whr th data ar of a multpl dampd snusodal natur, th man concrn hr s th stmaton of th dcay rats from th sampld data squnc. In ths papr, a mthod of stmatng th dcay rats and numbr of componnts of an xponntal sgnal s prsntd. Th mthod uss th Gardnr transform [] as modfd by Nchols t al [] to convrt th multxponntal sgnal nto a convoluton modl whch s dconvolvd usng a xponntal compnsaton dconvoluton (ECD [3]. Th rsultng nonstatonary dconvolvd data s statonarzd and furthr procssd usng VD-ARA as proposd by alam and dk [4]. Analyss of transnt multxponntal data usng Gardnr transform has bn don wth dffrnt modfcatons [5]. Th novlty n th work prsntd hr s th combnaton of ECD wth VD-ARA mthod whch s shown to outprform svral xstng mthods of multcomponnt transnt data analyss.. ECD/ARA Combnaton Applcaton of modfd Gardnr transform on th sgnal n Eq. ( was shown [6] to yld th followng nput dstrbuton functon whch contans paramtrs of ntrst 4

2 x( t = B δ ( t + ln λ ( = whr B. Th DFT of x (t ylds = ( λ α A Xˆ ( k = = B πk j ln λ N + ε ( k (3 whr ε (k s nonstatonary nos and k =,,...N d. Th nonstatonarty problm was addrssd n [7] usng a rctangular wndow and th rsultng squnc was: xˆ( k = = B πk j ln λ N + ( k whr (k s th nw statonary nos and k =,,...N d. N d = N +. Th dconvolvd and truncatd data can now b can b consdrd to b th output of an ARA modl whos nput s a complx wht nos squnc (k so that p n= q an x( k n = bn( k n, a = (5 n= whr a n and b n rprsnt rspctvly th AR and movng avrag (A modl coffcnts and p and q ar AR and A modl ordrs rspctvly. Th rmanng procdur s as dtald n [4]. Th guss valus of th AR and A modl ordr ar rspctvly p and q and th dsrd powr dstrbuton of x (t dnotd as P x (t s obtand as follows: 3. mulaton Rsults z= xp jπt NΔt P ( t = ( z = B δ ( t ln λ x In ordr to nvstgat th ffcacy of th proposd combnaton, smulatons wr carrd out and prsntd n ths scton. Frst, th truncaton pont N was stablshd usng th Cramr Rao Lowr Bound [7]. Two synthtc sgnals ar thn usd to stablsh th ffctvnss of th proposd tchnqu. pcfcally, what s nvstgatd hr s th ablty of th combnaton to analyz a smpl two-componnt sgnal and ts rsolvng powr whn hgh rsoluton sgnal s nvolvd. 3.. Dtrmnaton of th Truncaton Pont Th truncaton pont, N s crtcal to th prformanc of any tchnqu to b usd to procss th dconvolvd data, xˆ. CRLB bng a good masur of th ffcncy of paramtr stmats has bn usd to stablsh th qualty of th paramtr stmats for dffrnt valus of N. By varyng th data lngth and comparng of th rsultng stmats wth th CRLB w can know whch data lngth would produc th bst stmator. Drvaton of th CRLB was don as prsntd n [8]. Th sgnal usd for ths smulaton s ( = n( (7 Th rasons for th choc of ths sgnal wr gvn n [3] mulatons usng th proposd ARA algorthm wth convntonal nvrs fltrng showd that th spctrum s good only for 7 N 33 wth th bst prformanc at N 7. f k = = k k (4 (6 3.. Rsolvablty of th Componnts 4

3 To nvstgat th capablty of th proposd combnaton to analyz basc sgnals, th followng twocomponnt sgnal was usd:.. ( =. +. (8 Th dstrbuton functon s: x ( t =. δ ( t ln +. δ ( t ( α ( α Th sgnal was synthszd n ATLAB wth nos addd usng th functon awgn whch s an mbddd ATLAB functon. Tabl shows th rsult of applyng th proposd combnaton. ovr low ( NR 4dB, mdum ( 4 db < NR < db and hgh ( NR db NRs. A plot of th dstrbuton functon for hgh NR s shown n Fgur. Th dstrbuton functon was plottd aganst ngatv tm n ordr to arrang th xponnts n ascndng ordr on th horzontal axs. For th purpos of comparson, DFT plot s shown (brokn lns along wth th actual plot (sold. Th DFT graphs ar obtand by wndowng th dconvolvd data to rmov hgh frquncy nos and thn nvrs transformng, nstad of usng VD-ARA or any othr paramtrc modllng tchnqu. Th rsults ar basd on p =, q = 5 and th choc of dconvoluton paramtrs s accordng to Tabl. Tabl : Rcommndd dconvoluton paramtrs Paramtr NR Low dum Hgh a b 4 3 Tabl : Estmatd log of dcay rats ( ln for ( λ Expctd LOW EDIU HIGH Valu NR NR NR ln 5 ( Rspons to a hgh rsoluton sgnal Th followng sgnal was usd to tst th ffctvnss of th proposd combnaton n rsolvng a hgh rsoluton sgnal..5 5 ( = n( ( λ λ λ4 It s notworthy that for ths sgnal = = =. Ths sgnal would prov dffcult whn λ λ3 λ5 appld to most of th popular mthods (Prony, orgnal or modfd, nonlnar last squars, ET, tc.. Ths sgnal s thrfor sutabl to tst th ffctvnss of our proposd tchnqu. Th dstrbuton functon for th sgnal ( s as follows: 43

4 - Normalzd agntud (db Log (lambda Fg. : Powr dstrbuton for x ( at hgh NR usng th proposd combnaton. t ( α ( α ( α ( α δ ( t ln + δ ( t + δ ( t + ln + 5 δ ( t + ln 5 + δ ( t ln x ( t =.5 + ( Th rsult of applyng th proposd combnaton on ( s shown n Tabl 3. Fgur shows powr dstrbuton for x ( t at a mdum NR. It s obsrvd that whl th combnaton gvs good rsults ovr mdum and low NR, t ylds poor stmats at low NRs. Its dtcton thrshold s thus lowr than that of UIC (ultpl sgnal classfcaton and mnmum norm mthods usd n [3]. 4. Dscusson and Concluson In ths papr, a nw combnaton of a dconvoluton tchnqu and a paramtrc mthod has bn ntroducd. Th mthod has bn usd to analyz two sgnals, on wth two componnts and anothr wth fv componnts and hghr rsoluton. Although th combnaton rsults n a lowr dtcton thrshold than that of ECD/UIC and ECD/mnmum norm, t nvrthlss suprsds a fw arlr mthods lk Fast odl-fr Dconvoluton [9], Isrnbrg mthods of momnts (IO [] and ultxponntal transnt spctroscopy (ET [] n trms of th numbr of componnts t can b usd to analyz. Tabl 3: Estmatd log of dcay rats ( ln for ( λ 44

5 - Normalzd agntud (db Log (lambda 5. Rfrncs Fg. : Powr dstrbuton for x ( at mdum NR. [] D.G. Gardnr, J.C. Gardnr, G. Lush, and W.R. War. thod for th analyss of multcomponnt xponntal dcay curvs, Journ. Chm. Phys. 959, 3: [].T. Nchols,. R. mth, and. J. E. alam, Hgh Rsoluton Estmats n ultcomponnt gnal Analyss, Tch. Rport, Dpt. Elc. Engg., Unvrsty of Calgary, Alta, pt., 983. [3] A.U. Jba and.j.e. alam. Analyss of transnt multxponntal sgnals usng xponntal compnsaton dconvoluton. asurmnt 45( 9 9. [4].J.E. alam,.n. dk. Paramtr stmaton of multcomponnt transnt sgnals usng dconvoluton and ARA modlng tchnqus. Journal of chancal systms and sgnal procssng, 3, 7 (6 : - 8,. [5] A.U. Jba and.j.e. alam. An Apprasal of Gardnr Transform-Basd thods of Transnt ultxponntal gnal Analyss. Intrnatonal Journal of Computr Thory and Engnrng,, 4 (: 6-5. [6].J.E. alam. Applcaton of ARA modls n multcomponnt sgnal analyss. Ph.D. Dssrtaton, Dpt. of Elctrcal Engnrng, Unvrsty of Calgary, Calgary, Canada, 985. [7] A.U. Jba and.j.e. alam. Transnt ultxponntal Data lcton Usng Cramr-Rao Lowr Bound. Proc. Intrnatonal Confrnc on Computr and Automaton Engnrng, umba, January : 7-3. [8] A.U. Jba,.J.E. alam, O.O. Khalfa, F.A.. Elfak. Cramr-Rao Lowr Bound for Paramtr Estmaton of ultxponntal. Proc. of 6 th Intrnatonal Confrnc on ystms, gnals and Imag Procssng (IWIP 9, Chalkda, Grc. 9: -5. [9] J.A. Jo, T. Papaoannou, L. arcu, Fast modl-fr dconvoluton of fluorscnc dcay for analyss of bologcal systms. Journal of Bomdcal Optcs, 4, 9 (4: [] I. Isrnbrg, Robust stmaton n puls fluoromtry: a study of th mthod of momnts and last squars. Th Bophyscal Journal (: []. arco, J. amtr, J.R. orant, A novl tm-doman mthod to analys multcomponnt xponntal transnts, asurmnt cnc & Tchnology, 995, 6: t 45