Estimation of the Amplitude Quotient of Signals with Common Frequency. Tomaž Lušin 1, Dušan Agrež 2

Transcription

1 9 th Syposiu IMEKO TC Syposiu and th IWDC Workshop dvancs in Instruntation and Snsors Introprability July 8-9, 0, Barclona, Spain Estiation of th plitud Quotint of Signals with Coon Frquncy Toaž Lušin, Dušan grž Mtrl d.d., Ljubljanska csta, SI- Horjul, Slovnia, hon: , -ail: Univrsity of Ljubljana, Faculty of Elctrical Enginring, Trzaska, Ljubljana, Slovnia, phon: , fax , -ail: bstract - Th papr prsnts sipl singl-stp algoriths for fast asurnt and stiation of th aplitud ratio of th sinusoidal signals with th sa frquncy fro two channls. Th ida of th systatic rror rduction with th wightd aplitud DFT cofficints around th coponnt paks is usd in th coparativ asurnt procdur. apr analyss and copars th systatic bias rrors and nois rror bhaviors of th aplitud quotint stiation changing th ordr of Rif-Vincnt windows class I, which ar dsignd for axiization of th window spctru sid-lobs fall-off. Th inial bias rrors ar shown in rlation to th nubr of signal cycls in th asurnt intrval. K ywo rd s- plitud quotint, nonparatric stiation, frquncy doain, coparativ thod, Rif- Vincnt windows I. Introduction In th two channl acquisition systs, it is oftn to stiat th paratrs of two coon frquncy sinusoidal signals (lik in vry wid aras of ipdanc asurnts []-[], powr asurnts [], tc.. In th coparativ asurnts (ipdanc asurnt, rror valuation of transforrs [], tc. valus of th aplitud and phas of ach channl ar not rquird. Th only valus ndd ar in fact th aplitud ratio and th phas diffrnc of th invstigatd coponnts fro channl and with th sa frquncy f. Sapling by frquncy t g t coposd of M coponnts (with f,, and f s of th priodic band liitd analog signal as frquncy, aplitud, and phas, rspctivly can b xprss as follows: w M 0 nt gnt wnt sin f nt n0,..., N ( To stiat paratrs of th ti dpndnt signals, containing any priodicity, it is ost suitabl to us a transforation of th signal in th frquncy doain. Th discrt Fourir transforation (DFT of th w k g k on N sapld points ( at th spctral lin i is givn by: windowd signal M j Wi Wi i j 0 -j ( whr f f i is th coponnt frquncy rlatd to bas frquncy rsolution f N t and consists of an intgr part and th non-cohrnt sapling displacnt tr. Evaluations of th ti-discrt spctra, howvr, ar haprd by th window lakag, which occurs if a nonintgr nubr of priods ar prsntd in th sapld data st and this is th usual situation in ral asurnts. Th lakag ffcts can b vry rducd with th coparativ asurnt, if th siultanousnss of th sapling on both channls is assud and th asurnt ti of signals is th sa. Th assuranc of ths conditions givs qual displacnts. In this papr w try to show th ffctivnss of th systatic rror rduction using wightd aplitud DFT cofficints of two sin signals with coon frquncy. In stiations th Rif-Vincnt windows class I hav bn usd. II. Estiation of th aplitud quotint Th DFT cofficints surrounding on signal coponnt ar du to th short-rang lakag contribution of th window spctru wightd by th aplitud of th coponnt (fro th first tr in ( and th long-rang lakag contributions: fro th scond tr in ( of th invstigatd coponnt and fro both trs in ( of othr coponnts. ISBN-0: ISBN-:

2 9 th Syposiu IMEKO TC Syposiu and th IWDC Workshop dvancs in Instruntation and Snsors Introprability July 8-9, 0, Barclona, Spain j j i W Wi ik M ( k0, k For on significant coponnt only, th largst local aplitud DFT cofficint can b xprssd with th shortrang lakag contribution and fro and th long-rang lakag contributions collctd in W hr, th aplitud can b dducd: i W i Th sa xprssion is valid for aplituds of both channls, i W i i ( and, rspctivly. Looking for th quotint of th aplituds of two signals with th sa frquncy, th window function W in th dnoinator can b canclld. Nglcting th long-rang lakag contributions at on DFT cofficint i i, th quotint of th aplituds can b stiatd by th quotint of th axial local aplitud DFT cofficints of two signals.,, i i i i,, i i ( Th long-rang lakag contributions can b rducd in thr ways: by using windows with fastr dcrasing of th sid lobs than th rctangular window (lik Rif-Vincnt windows, Dolph-Chbychv windows, tc., or by using th ulti-point intrpolatd DFT algoriths, or by iniizing th phas diffrnc of th coponnts of th two signals 0 [].,, In th sa annr as in (, w can obtain th aplitud of th signal by suing th largst two local DFT cofficints around th signal coponnt (s is th sign of displacnt [6]. Following th fact that th succssiv lakags changing thir sign owing to th asurnt ti truncation and consquntly having th sinc function in th window krnl, th long-rang lakags i s i i i can b rducd by suation of th largst local aplitud DFT cofficints: i i s W i W i s W W i, (6 i i s i W W ( Th dnoinator in (, which is dfind by th ployd window, can b canclld in th quotint:,, i i s i i i s i,, i i s i i s (8 Th singl aplitud can b stiatd with th thr largst local aplitud cofficints [6]: i i i i W W W (9 Th window dpndnt dnoinator and th short-lakag window contributions in nurator i i i i i can b cancld in th quotint:, i i i i i i,. (0 W can us th sa procdur with th wightd fiv largst cofficints and intrpolation. In th last tr of or th quation ( th absolut valu of th diffrnc is usd whn on of th cofficints i drops out of th spctru ain lob ( f i wid in th Hann window cas and gts th ngativ sign. ISBN-0: ISBN-:

3 9 th Syposiu IMEKO TC Syposiu and th IWDC Workshop dvancs in Instruntation and Snsors Introprability July 8-9, 0, Barclona, Spain,, i i i i i i i i i i 6 ( 6 Th sa procdur can b usd also with th svn largst local aplitud cofficints [6]:,, i i i i 6 i i i i i i i i i i ( Fixing th ti, th rduction of th systatic lakag rror ostly dpnds on th usd window. For th sak of analytical siplicity cosin-class windows ar frquntly usd []. Windows of th class I (RV-I ar dsignd for axiization window spctru sid-lobs fall-off basd on a nubr of th ti doain window drivativs zros at th window nds [8],[9]: wn DI, l cos l n, n 0,,..., N ( l0 N whr D I, l ar th wightd cofficints of cosins in th window function. nubr of th usd cosins functions dfins window ordr. Whn th ordr is 0, th cofficint D I, 0 is and th quation (8 givs th rctangular shap. If is w gt th Hann window. Highr valus of xpand th window transfor ainlob and rduc th spctral lakag (Fig.. Th wightd cofficints can b calculatd using xprssion l C y C l C x x! x y! y! : DI,0 ; D I, l ; l,,...,. Frquncy spctru can b gnrally xprssd by [8]: W! sinπ π N N - j - j N N W l l ( 0 0 W W in(rv-i( Figur. Th noralizd spctra shaps of th RV-I windows: 0 th rctangular window, th Hann window,. Using suitabl ordr of th RV-I windows in dpndnc of th rlativ frquncy will giv th lowst systatic lakag rror in th paratrs stiations. W chckd th rror of th aplitud quotint * stiation ( * is th tru valu of th aplitud quotint for on sin coponnt in both signals g t and t =0 = = = = g with a doubl scan varying both coon frquncy and th phas of th scond signal bcaus th long-rang lakags ar frquncy and phas dpndnt ( ; N 0 ; 0. 6, 0. 0 and 0, 90 90,. Th absolut axiu valus of rrors (fro itrations at th givn rlativ frquncy wr copard for th RV-I windows with diffrnt ordrs 0 (Fig. : on-point stiation by (, thr-point stiation by (0. ISBN-0: ISBN-:

4 9 th Syposiu IMEKO TC Syposiu and th IWDC Workshop dvancs in Instruntation and Snsors Introprability July 8-9, 0, Barclona, Spain ax ( ax ( =0 = = = = 8 0 in ax (, RV I = = = = = 8 0 in ax (, RV I = a th on-point stiation by ( b th thr-point stiation by (0 Figur. Maxial rlativ valus of rrors of th aplitud quotint stiation using RV-I windows and th curv of th inial axial rrors dpndning on th corrsponding window ordr Coparing th bias systatic rrors (Figs. and shows that RV-I windows prfor widr stiation rror ain-lob with incrasing th ordr and ffctivly rduc th long-rang rrors with incrasing frquncy. ax ( =0 = = = = 8 0 in ax (, RV I = Figur. Maxial rlativ valus of rrors of th aplitud quotint stiation with th two-point intrpolation using RV-I windows by (8 and th curv of th inial axial rrors dpndning on th corrsponding window ordr Looking for th bst prforanc, th inial curv of th lakag rrors could b found. Fig. shows collctd inial curvs of th axial bias rrors of th aplitud quotint stiations with th on-, two-, thr-, fivand svn-point intrpolations of th DFT using RV-I windows and th corrsponding ordrs..., with which ths inial valus hav bn obtaind. Th odd nubrs of intrpolation DFT points (p, p,.. prfor th sa inial curvs of th axial rrors of th aplitud quotint stiations but th RV-I window ordr has to b adaptd in rlation to signal cycls in th asurnt intrval and th nubr of intrpolation points. In th vicinity of th lowst systatic bias rrors ar achivd by th p stiation and ordr or cobination p. In th vicinity of, this is achivd by cobinations: ISBN-0: ISBN-:

5 9 th Syposiu IMEKO TC Syposiu and th IWDC Workshop dvancs in Instruntation and Snsors Introprability July 8-9, 0, Barclona, Spain p or p or p, in th vicinity of, this is achivd by following cobinations: p or p or p or p, tc. To achiv th lowst systatic bias rrors w can incras th nubr of th usd DFT cofficints in stiations and dcras th ordr of th RV-I windows or fix th nubr of th usd DFT cofficints (typically thr cofficints and incras th ordr (Fig.. Th vn nubrs of intrpolation points prfor th sa bhavior (Figs. and. 0 0 ax ( in ax (, RV I,,, in ax (, RV I a b c Figur. Minial curvs of th axial rrors of th aplitud quotint stiations with intrpolations of th DFT using RV-I windows and th corsponding ordrs... : a - th on-point st., b - th thr-point st., c - th fiv-point st., d - th svn-point stiation Th rduction of th systatic rrors incrass th contribution of th nois rando part of rrors [6]. Th quantization rror is a iniu that has to b takn into considration in th asurnt uncrtainty of th final rsult. In siulations, th whit nois with th rctangular distribution, zro an and th standard dviation t nois was addd to both signals in th ti doain. t vry tst point with changing frquncy and phas lik in Fig, 0 trials of rando addd nois wr usd for th stiation of th quotint standard dviation (Figs. and 6: at vry frquncy altogthr trials. Th corrsponding signal to nois ratio in th ti doain for th particular signal was SNR i i i. Th nois propagations in th algoriths wr copard to th Craér-Rao lowr bound [0]. SNR SNR ( N SNR SNR d ( ( = = = = =0 6 = = = = =0 a th on-point stiation by ( b th thr-point stiation by (0 Figur. Standar dviations of rrors of th aplitud quotint stiation using RV-I windows ; 6 signal and :, nois, nois, 0. 00, SNR SNR, 0 ( 6,dB ; 6 N 0 ; 6 0 ISBN-0: ISBN-:

6 9 th Syposiu IMEKO TC Syposiu and th IWDC Workshop dvancs in Instruntation and Snsors Introprability July 8-9, 0, Barclona, Spain Coparison of th stiation nois rrors (Fig. shows that w hav th sa lvl of standard dviations with or p and p stiations xcpt that with th last on th ordr is lowr for on: p p p p or, tc. Siulations show (Fig. 6 that w can gnraliz this qual bhaviors p p p p p p p p, tc. In paralll to systatic rror, w can conclud that th lowst systatic bias rrors at particular rlativ frquncy hav th sa nois rror lvl (for xapl: in th vicinity of, inial bias is achivd by cobinations: p or p or p (Fig. but nois lvl is th sa (Figs. and 6. p p p ( ( = = = = = = 6 a th fiv-point intrpolation by ( b th svn-point intrpolation by ( Figur 6. Standar dviations of rrors of th aplitud quotint stiation using RV-I windows I. Conclusions Singl-stp algoriths for fast asurnt and stiation of th aplitud ratio of th sinusoidal signals with th sa frquncy fro two channls ar prsntd. aratrs ar calculatd fro th DFT cofficints around th coponnt paks. apr analyss and copars th systatic bias rror and th nois rror bhaviors changing th ordr of Rif-Vincnt windows class I. Th lowst systatic bias rrors can b found with incrasing th ordr of RV-I windows whn a nubr of cycls in th asurnt ti incrass. Th nois propagations using highr ordr widows ar about.- to.-tis largr than th CR lowr bound and ar displacnt dpndnt. REFERENCES []. M Raos,. C. Srra, " nw sin-fitting algorith for accurat aplitud and phas asurnts in two channl acquisition ssysts", Masurnt, vol., no., pp. -, Fbruary 008. [] L. grisani,. Baccigalupi,. itrosanto, digital signal-procssing instrunt for ipdanc asurnt, IEEE Trans. Instru. Mas., vol., no. 6, pp. 90-9, Dcbr 996. []. Carbon, D. tri, vrag owr Estiation undr Nonsinusoidal Conditions, IEEE Trans. Instru. Mas., vol. 9, pp. -6, pril 000. []. Cataliotti, D. Di Cara,. E. Eanul, S. Nuccio, "Currnt Transforrs Effcts on th Masurnt of Haronic ctiv owr in LV and MV Ntworks", IEEE Trans. on owr Dlivry, vol. 6, no., pp , January 0. [] D. grž, "Estiation of paratrs of two sin signals with coon frquncy", rocdings of th IEEE I MTC/00, ustin (Tx, US, pp. 6-, May -6, 00. [6] D. grž, Wightd Multi-oint Intrpolatd DFT to Iprov plitud Estiation of Multi-Frquncy Signal, IEEE Trans. Instru. Mas., vol., no., pp. 8-9, pril 00. []. ndria, M. Savino,. Trotta, "Windows and intrpolation algoriths to iprov lctrical asurnt accuracy", IEEE Trans. Instru. Mas., vol. 8, no. 8, pp , ugust 989. [8] D. Blga, D. Dallt, Multifrquncy signal analysis by intrpolatd DFT thod with axiu sidlob dcay windows, Masurnt, Vol., No., pp. 0-6, pril 009. [9] M. Sdláčk, Z. Stoudk, Dsign of DS windows using window spctru zros placnt, roc. XIX IMEKO World Congrss, pp. 6-69, Lisabon, ortugal, Sptbr 009. [0]. Händl, "aratr stiation ploying a dual-channl sin-wav odl undr a aussian assuption", IEEE Trans. Instru. Mas., vol., no. 8, pp , ugust 008. ISBN-0: ISBN-: