Two New Sliding DTFT Algorithms for Phase Difference Measurement Based on a New Kind of Windows

Transcription

1 .78/sr--8 MEASUEMET SCIECE EVIEW, Volu, o. 6, Two w Sliding DTFT Algoriths for Phas Diffrnc Masurnt Basd on a w Kind of Windows Yaqing Tu, Ting ao Shn,, Haitao Zhang, Ming Li Dpartnt of Inforation Enginring, Logistical Enginring Univrsity, Chongqing, China Corrsponding author: poplsta@sina.co For th ultra-low frquncy signals or adjacnt yquist frquncy signals, which xist in th vibration nginring doain, th traditional DTFT-basd algorith shows srious bias for phas diffrnc asurnt. It is indicatd that th spctru lakag and ngativ frquncy contribution ar th ssntial causs of th bias. In ordr to iprov th phas diffrnc asurnt accuracy of th DTFT-basd algorith, two nw sliding DTFT algoriths for phas diffrnc asurnt basd on a nw kind of windows ar proposd, rspctivly. Firstly, th nw kind of windows dvlopd by convolving convntional rctangular windows is introducd, which obtains a strongr inhibition of spctru lakag. Thn, with ngativ frquncy contribution considrd, two nw forulas for phas diffrnc calculation undr th nw kind of windows ar drivd in dtail. Finally, th ida of sliding rcursiv is proposd to dcras th coputational load. Th proposd algoriths ar asy to b ralizd and hav a highr accuracy than th traditional DTFT-basd algorith. Siulations and nginring applications vrifid th fasibility and ffctivnss of th proposd algoriths. Kywords: Phas diffrnc, discrt ti Fourir transfor, cosin window, spctral lakag, ngativ frquncy. T. ITODUCTIO HE POBLEMS of stiating th phas diffrnc btwn signals rcivd at two sparatd snsors ar considrd in any aras such as fault diagnosis, dirction finding, and sourc localization. Many thods hav bn proposd for phas diffrnc asurnt in th past thr dcads [-[6. Gnralizd Cross Corrlator (GCC) [7-[9 is a convntional approach to stiat th phas diffrnc by locating th cross corrlation pak of th filtrd vrsion of two rcivd signals, and it has bn provn that optiu prforanc can b attaind whn th signals and noiss ar Gaussian distributd. Howvr, th thod rquirs a prior statistics of th rcivd signals and it fails to work whn th noiss ar ipulsiv or spatially corrlatd. Whn th sourc signals ar dtrinistic, a Discrt-ti Fourir transfor (DTFT)-basd algorith is proposd for singl coplx sinusoids [, whil a Quadratur Dlay Estiator algorith is dvlopd for ral-valud sinusoids by utilizing th in-phas and quadratur-phas coponnts of on of th rcivd signals [. Using th ida of [, two odifid thods hav bn dvlopd for ral sinusoidal signals [. Th first thod is rfrrd to as UQDE, which rovs th bias of QDE by utilizing all th in-phas and quadratur-phas coponnts of th rcivd signals. Th scond thod is rfrrd to as odifid DTFT algorith, which calculats th phas diffrnc of DTFTs of two coplx sinusoids drivd fro th ral signals. Th ky ida of th odifid DTFT algorith is to transfor th ral ton to a coplx on with th known frquncy inforation, and this is th diffrnc btwn th odifid DTFT algorith and th typical DTFT-basd algorith. Howvr, all th algoriths ar difficult to achiv in practic if th frquncy is unknown. Aong th phas diffrnc asurnt thods for ral signals with unknown frquncy, th Discrt ti Fourir transfor-basd algorith is usd as th typical on. Th DTFT-basd algorith calculats th phas diffrnc of DTFTs of two ral sinusoids at th stiatd signal frquncy. Howvr, th algorith nglcts th contribution of ngativ frquncy for ral-valud sinusoids. Whn th signal frquncy is quit low or clos to th yquist frquncy, th algorith brings about significant bias or vn bcos inffctiv. Th sa thing occurs whn th availabl sapld data for DTFT calculation ar not nough. To rov th bias and iprov th accuracy of th DTFT-basd algorith, two nw sliding DTFT algoriths ar proposd in this papr. In Sction, th proposd algoriths and th procss of thir application ar dscribd in dtail, which introduc th nw kind of windows, th windowd DTFT algoriths with ngativ frquncy contribution and th ida of sliding rcursiv, rspctivly. In Sction, th proposd algoriths ar validatd by siulations and xprints. Finally, Sction prsnts th conclusion.. THE PICIPLE OF THE POPOSED ALGOITHMS AD THEI APPLICATIO POCESS A. A nw kind of windows and its charactristic. A nw kind of windows, calld convolvd-windows, which is dvlopd by convolving convntional rctangular window, was introducd in [. This kind of windows has th advantags of sipl structur, lowr attnuation and good inhibition of spctral lakag. Th construction of th convolvd-windows can b xprssd as follows. Considr th rctangular window ti doain and frquncy doain xprssions as w ( ) n =, n M () M ( ) j j sin M / W ( ) = () sin ( / ) 5

2 MEASUEMET SCIECE EVIEW, Volu, o. 6, whr M is th lngth of rctangular window, and is th digital signal frquncy. A nw M - data squnc can b obtaind by convolving rctangular windows. If zro is addd in th front or back of th nw squnc, a nw M data squnc can b obtaind, calld -ordr rctangular slf-convolution window (SCW). Add zro in th front of th M - data squnc and considr = M, th ti doain and frquncy doain xprssions of -ordr SCW can b xprssd as n n / w ( n) = n / n () ( ) j j sin / W ( ) = () sin ( / ) Siilarly, M data squnc can b obtaind by convolving rctangular windows. A nw -ordr SCW can b dvlopd by adding zro in th front of th nw squnc and zro at th back of th nw squnc. Considr = M, th ti doain and frquncy doain xprssions can b xprssd as w n n n 6 6 n n n n ( n) = 5 n n n n 8 96 n n n n (5) ( 8) j j sin / W ( ) = (6) sin ( / ) According to th construction of th SCW, th -ordr SCW can b dvlopd in a siilar way. At first, a nw data squnc can b obtaind by convolving rctangular windows, and thn, if is vn, th -ordr SCW can b dvlopd by adding / zro in th front of th nw squnc and ( ) / zro at th back of th nw squnc, whil is odd, th -ordr SCW can b dvlopd by adding ( ) / zro in th front of th nw squnc and ( ) / zro at th back of th nw squnc, and thn, th ti doain and frquncy doain xprssions of -ordr SCW ( = M ) can b xprssd as sin ( / ) j if = k j sin ( / ) W ( ) = k Z sin ( / ) j if = k sin ( / ) (7) It is obviously fro forula (7) that th width of th ain lob bcos largr with th incras of, accordingly, th ordr of SCW is usually lss than. In addition, in th zro points of its aplitud-frquncy charactristic, th valu of - ordr drivativs is zro. As a rsult of this charactr, th intrfrncs btwn th haronics du to spctru lakag can b rducd furthst by applying th SCW. Consquntly, th prcision of haronics stiation can b boostd. B. Th windowd DTFT algoriths. ) Th asurnt principl of DTFT algorith Considr two ral sinusoids with th sa frquncy x( n) = A cos( n θ),,, L, (8) x ( n) = A cos( n θ ) whr A and A ar aplituds, θ and θ ar initial phass. Mark as th stiatd valu of, th DTFT of x ) at can b coputd as [ (, n j n (,) ( ) =, cos( θ, ) A, j( n θ, ) j( n θ, ) j n [ = X A n According to Eulr s forula, a ral sinusoid can b forulatd as th su of two xponntial signals with positiv and ngativ frquncis, rspctivly. If th ngativ frquncis in (9) ar nglctd, thn A X ( ) = (,), ( θ, ) j n j n A = A ( ), j[ θ,, Th phas of ( ) ϕ and as θ = θ -θ (9) ( ) sin( ), if sin( ) jθ,, if = X and ( ) () X is dnotd by ϕ, rspctivly. If th phas diffrnc is dfind, it can b calculatd as 5

3 MEASUEMET SCIECE EVIEW, Volu, o. 6, θ = ϕ ϕ () This is th typical DTFT-basd algorith. It is obvious that th phas diffrnc of two signals approxiats th subtraction of two DTFT phass at th stiatd signal frquncy. If th ngativ frquncy is nglctd in (), it is no longr th DTFT of th ral sinusoidal signal but is that of th coplx on. In othr words, th contribution of ngativ frquncy coponnts in th spctru is nglctd in typical DTFT-basd algorith. Whn th signal frquncy is quit low or clos to th yquist frquncy, th ngativ frquncy intrfrnc in th spctru bcos rarkabl, which will bring about significant bias in phas diffrnc asurnt. Th sa thing will occur if a sall nubr of sapld data ar takn in DTFT calculation. Thus, two nw DTFT algoriths with ngativ frquncy contribution considrd, which ai to rov th bias of th DTFT-basd algorith, ar prsntd. Two nw forulas for phas diffrnc asurnt adopting -ordr SCW and -ordr SCW ar drivd, rspctivly. ) Th nw DTFT algorith basd on -ordr SCW Assuing th lngth of -ordr SCW is, ultiply th -ordr SCW with signals x ), and th DTFT of th signals can b xprssd as, ( n X ( ) = A, cos( n θ,(,), ) w ( n) jn A, j( n θ, ) j( n θ, ) = [ w ( n) jn A, jθ [, jθ = W, ( ) W ( ) ( ) sin ( ) ( A ), j[ θ, = sin ( ) ( ) A sin ( ) ( ), j[ θ, sin ( ) () As is gnrally known, ultiplying in ti doain is in corrspondnc with convolving in frquncy doain. So th quation can b rgardd as X ) = X ( ) W ( ) ( (, ),(,) Assuing that and thn,, () can b drivd fro (). I[ X ( ) c c,(,),(,) = =,(,) [ X ( ) c,(,) c,(,) c c (),(,) = (),(,) c c,(,) whr c ϕ dnots th phas of X ( ).,(, ) = sin a sin a cosa sin a sin a cos c c = sin a sin a sina sin a sin a sina = sin a sin a sina sin a sin a sina c = sin a sin a cosa sin a sin a cosa a = ( )/, a = ( )/, a = ( )/, a = ( )/ Making us of th forula tan( θ, θ, can b dducd as:, ) =,,,,(,) a, th phas diffrnc ( cc c c )(tan ϕ tan ϕ ) θ= arctan[ ( c c ) ( c c cc )(tan tan ) ( c c )tan tan,, ϕ ϕ ϕ ϕ,,,, (5) Whn th signal-to-nois ratio (S) of signals is not vry low, th signal frquncy stiatd by adaptiv notch filtr or discrt spctru corrction is gnrally quit clos to th tru valu, i..,, thn, sina /sina / can b dducd. And thn, quation 5 can approxiatly b xprssd as follows. ( ),, θ = arctan [ ( ) whr, = ( sin ) /6 sin ;,, = ( sin ) /6 sin [( sin sin) cos / ; = [( sin sin) sin / ; = ( sin ) /6 sin [( sin sin) cos /; = /, (6) Equation 6 is th nw DTFT algorith basd on -ordr SCW for phas diffrnc calculation. Whn =, th forula for phas diffrnc calculation is just th sa as (6), so thr is no nd to stiat if th quals or not. Howvr, th highr accuracy of frquncy stiation, th bttr th phas diffrnc asurnt. ) Th nw DTFT algorith basd on -ordr SCW Th ti doain and frquncy doain of -ordr SCW xprssions ar illustratd in quation 5 and 6, rspctivly. According to th drivation procss of th nw DTFT algorith basd on -ordr SCW for phas diffrnc asurnt, th nw DTFT algorith basd on -ordr SCW for phas diffrnc calculation can b obtaind siilarly as 5

4 MEASUEMET SCIECE EVIEW, Volu, o. 6, Whr 5 ( tan ϕ ),, θ= arctan [ ( tan ϕ ) = ( sin / ) sin ; 6 7 8,,,, 8 8 = ( sin / ) sin [( sin sin / ) cos ; 6 = [( sin sin / ) sin ; = ( sin /) sin [( sin sin /) cos ; 8 = / C. Th sliding rcursiv DTFT algorith. (7) If th signals ar ti-varying, th algorith of DTFT cannot b carrid out dirctly. As th width valu of window is vry sall and th frquncy varis a bit in th window which can b approxiatly rckond, th frquncy dos not chang in th window, and thn, th phas diffrnc of ach sapld datu can b workd out basd on th sliding intrcpting thought. For signal x( n ), assu that data hav bn sapld at th ti of, i.. x( ), x( ) x( ), Th DTFT of th sapling squnc at can b xprssd as X, j n x( n) ( ) = = x( ) x( )* x( )* L x( )* j j j ( ) (8) At th ti of, a nw datu x( ) is sapld and addd to th data, whil th x( ) is liinatd fro th data, th DTFT of th squnc with data at can b xprssd as X, ( ) = x( n) j n = x( ) x( )* L x( )* x( )* j j ( ) j ( ) Fig.. Th sliding ti window for points. (9) Coparing (8) and (9), w obsrv that and dlgat th stiatd frquncy of two adjacnt window sapling points, rspctivly. It is obvious fro Fig.. that it is inaccurat to us to calculat th X, ( ), as th - points ar th sa in th two adjacnt windows. So, (9) can b rvisd as follows X j j ( ) j ( ), ( ) = x( ) x( )* L x( )* x( )* j n j j ( ) x( n) x( ) * x( )* = [ j n j j ( ) x( n) x( ) * x( )* [ = [ X ( ) * j j, x( ) x( )* () Equation is th sliding rcursiv DTFT algorith proposd in this papr. It is obvious that thr is a rcursiv rlationship xit aong th nw sapling squnc with th old on. Bsids, only coplx additions and coplx ultiplications ar rquird to calculat th nw DTFT, and th routin DTFT calculation is just ndd at th first rctangular window in th whol procss, which discards th rdundancy calculations. ot that th quation is quit suitabl for th signal whn th frquncy is stady. For ti-varying signal, if all which xist in th first undrlin part in () ar displacd by, th dynaic charactristic will b waknd, spcially if th frquncy varis in a big rang. As for th typical DTFT-basd algorith, th routin DTFT calculation is ndd in vry window without xcption. Whn a nw datu is obtaind, coplx ultiplications and - coplx additions ar rquird to calculat th nw DTFT. Bsids, according to Fig.., th convntional DTFT-basd algorith has - rdundancy calculations. D. Th stps of th proposd algorith. To su up, th stps of th proposd algoriths can b illuinatd as follows () Estiat th signal frquncy by adaptiv notch filtr or discrt spctru corrction, dnotd by i ; () Calculat th DTFTs by (9) at th first rctangular window at ; () Calculat and by convolving th -ordr SCW or -ordr SCW with DTFT in th frquncy doain; () Calculat to or 5 to 8 by and, and thn substitut into (6) or (7) for and to calculat th phas diffrnc; (5) Mak us of th sliding rcursiv DTFT algorith in ; (), calculat th DTFT at vry frquncy point i (6) pat stp () and stp (), th phas diffrnc can b obtaind at any ti. 5

5 MEASUEMET SCIECE EVIEW, Volu, o. 6,. SIMULATIO AD EXPEIMETAL ESULTS In ordr to validat th ffctivnss of th proposd thod, coputr siulations hav bn carrid out firstly. To draw a coparison, th phas diffrnc stiats ar givn at diffrnt conditions, such as undr noisy or noislss circustanc, undr diffrnt sapling data circustanc, and so on. Thn, xprintal data undr diffrnt flow rats is gathrd through th Coriolis ass flowtr platfor dsignd by our rsarch ta, which is usd for xprints. In siulations, th phas diffrnc quals.8º, th sapling frquncy quals Hz, th nubr of sapld points quals, and th frquncy rsolution quals.9766 Hz. A. Siulation rsults. Undr noislss circustancs, th rlativ rrors of phas diffrnc which ar coputd by coparing th stiatd valus of phas diffrnc vrsus th thortic valus ar shown in Fig.. and Fig.., rspctivly. Th signal frquncy varis fro Hz to Hz in Fig.., and varis fro 9 Hz to 99 Hz in Fig.., with th stp lngth of. Hz. As shown in Fig.. and Fig.., th DTFT-basd algorith causs significant rrors, whil th accuracy of th proposd algoriths is quit low, and thy ar always suprior to th DTFT-basd algorith in th absnc of nois. This is bcaus th sid lob of ngativ frquncy coponnts is considrd in th proposd algoriths. Bsids, th accuracy of th proposd algorith with th -ordr SCW is suprior to th on with -ordr SCW, for th -ordr SCW has a lowr attnuation and bttr inhibition of spctral lakag than -ordr SCW. lativ rrors of phas diffrnc (%) DTFT-basd algorith proposd algorith with -ordr SCW proposd algorith with -ordr SCW Fig.. lativ rrors of phas diffrnc in th absnc of nois whn th signal frquncy is quit low. For ral-valud signals in th prsnc of whit Gaussian nois, xtnsiv coputr siulations hav bn conductd to valuat th prforanc of th proposd algoriths. Th whit Gaussian noiss iposd on two sinusoids ar not corrlativ. Coparisons ar also ad by th DTFT-basd algorith with th proposd algoriths. All siulation rsults providd ar th avrag of indpndnt runs. lativ rrors of phas diffrnc (%) DTFT-basd algorith proposd algorith with -ordr SCW proposd algorith with -ordr SCW Fig.. lativ rrors of phas diffrnc in th absnc of nois whn th signal frquncy is clos to yquist frquncy. Fig.. and Fig.5. show th MSE of th proposd algoriths and DTFT-basd algorith vrsus th signal frquncy at S= db, rspctivly. It is obsrvd fro Fig.. and Fig.5. that th proposd algoriths ar alost unbiasd, whil th DTFT-basd algorith rsults in a significant bias whn th signal frquncy is quit low or clos to th yquist frquncy. Th bias of DTFT-basd algorith varis in th annr of dcaying oscillation, bcaus th sid lob of ngativ frquncy coponnts in th spctru quals to zro, whn signal frquncy quals to a ultipl of half a frquncy rsolution. In othr words, th contribution of ngativ frquncy can b ignord at this ont. What is or, th MSE of th DTFT-basd algorith approach thos of th proposd algoriths asyptotically whn th signal frquncy ovs away fro zro and yquist frquncy. Man squar rrors of phas diffrnc ( ) DTFT-basd algorith proposd algorith with -ordr SCW proposd algorith with -ordr SCW.5.5 Fig.. MSE of phas diffrnc undr S= db whn th signal frquncy is quit low. Fig.6. shows th MSE of th proposd algoriths, and DTFT-basd algorith vrsus th diffrnt sapling points varis fro to whn signal frquncy quals.5 Hz. It is sn that th DTFT-basd algorith approachs thos of th proposd algoriths asyptotically whn th signal frquncy ovs away fro zro and yquist frquncy, bsids, th biass of DTFT-basd algorith vary in th annr of dcaying oscillation. Both of ths rasons ar illuinatd as shown in Fig.. and Fig.5. 5

6 MEASUEMET SCIECE EVIEW, Volu, o. 6, Man squar rrors of phas diffrnc ( ) DTFT-basd algorith proposd algorith with -ordr SCW proposd algorith with -ordr SCW Fig.5. MSE of phas diffrnc undr S= db whn th signal frquncy is clos to yquist frquncy. causs significant rrors with th incras of sapling points, bcaus th proposd algorith adopts th sliding rcursiv ida, which can liinat th rdundancy calculation of DTFT. Th proposd algorith is of lss coputr load and has bttr dynaic prforanc. phas diffrnc ( ).5.5 thortic valu DTFT-basd algorith proposd algorith with -ordr SCW Man squar rrors of phas diffrnc ( ) DTFT-basd algorith proposd algorith with -ordr SCW proposd algorith with -ordr SCW Sapling points () Fig.6. MSE of phas diffrnc undr diffrnt sapling points. Fig.7. shows th coparison of phas diffrnc vrsus th sapling points stiatd by th DTFT-basd algorith and th proposd algorith with -ordr SCW. W just prsnt th proposd algorith with -ordr SCW, for th accuracy of th proposd algorith with -ordr SCW is quit clos to th proposd algorith with -ordr SCW, and th accuracy of th proposd algorith with th -ordr SCW is suprior to th on with -ordr SCW. As is shown in Fig.7., th proposd algorith can track th phas diffrnc continuously, whil th DTFT-basd algorith Sapling points ( ) x Fig.7. Th coparison curv of phas diffrnc stiatd by two algoriths. B.. Exprintal rsults. In ordr to discuss th ffctivnss in practic, aking us of th CMF plant which our group dvlopd to acquir larg nubrs of data, thn th DTFT-basd algorith and th proposd algorith with -ordr SCW ar valuatd. For CMF, th ass flow rat is calculatd by asuring th phas diffrnc or ti intrval btwn two signals dtctd by lctroagntic snsors. W slctd th HEOIK CMF with a HE8 transittr in th xprints, th rang of th ass flow rat varis fro. kg/in to 6.8 kg/in, and th signal frquncy of CMF approxiats 6 Hz. At ach stady flow rat, sapling points of data ar sapld ach ti for ach snsor. Tabl. shows th stiatd ti dlays and rlativ rrors undr diffrnt flow rats. As shown in Tabl., th rsults of th proposd algorith ar uch closr to th thortic valus of th ti dlays by coparison, whras th DTFT-basd algorith causs rarkabl dviations, which also validat th ffctivnss of th proposd algorith. Tabl. Th stiatd ti dlays and rlativ rrors undr diffrnt flow rats. Mass flow rat /(kg/in) Th thortic ti dlay valus/µs Th an of ti dlay valus/µs DTFT-basd algorith Th proposd algorith with -ordr SCW Th rlativ rror of ti dlay valus/% DTFT-basd algorith Th proposd algorith with -ordr SCW

7 MEASUEMET SCIECE EVIEW, Volu, o. 6,. COCLUSIO Th DTFT-basd algorith for phas diffrnc asurnt is biasd whn th frquncy is quit low or clos to th yquist frquncy. In ordr to iprov th accuracy of th DTFT-basd algorith, two nw sliding DTFT algoriths for phas diffrnc asurnt basd on a nw kind of windows hav bn dvlopd, and th phas diffrnc calculation forulas with -ordr SCW and -ordr SCW ar prsntd, rspctivly. Th proposd algoriths considring th ngativ frquncy contribution, adopting th SCW and th sliding rcursiv ida, can rov th bias of th DTFT-basd algorith and attain optiu prforanc all th ti, spcially for th proposd algorith with -ordr SCW. Siulations and xprintal rsults validat th ffctivnss of th proposd algoriths. For futur rsarch, w will focus on xtnding and gnralizing this typ of algoriths to a or gnral syst idntification sch, and furthr rsarch is undr discussion. ACKOWLEDGMET This work was supportd by th ational atural Scinc Foundation of China (679, 675) and th atural Scinc Foundation Projct of Chongqing (jja6, jcyja). EFEECES [ Ettr, D., Strarns, S. (98). Adaptiv stiation of ti dlays in sapld data systs. IEEE Transactions on Acoustics, Spch and Signal Procssing, 9 (), [ Maskll, D.L., Woods, G.S. (999). Th stiation of subsapl ti dlay of arrival in th discrt-ti asurnt of phas dlay. IEEE Transactions on Instruntation and Masurnt, 8 (6), 7-. [ Tu, Y.Q., Zhang, H.T. (8). Mthod for CMF signal procssing basd on th rcursiv DTFT algorith with ngativ frquncy contribution. IEEE Transactions on Instruntation and Masurnt, 57 (), [ aos, P.M., Cruz Srra, A. (8). A nw sin-fitting algorith for accurat aplitud and phas asurnts in two channl acquisition systs. Masurnt, (), 5-. [5 Vucijak,.M., Saranovac, L.V. (). A sipl algorith for th stiation of phas diffrnc btwn two sinusoidal voltags. IEEE Transactions on Instruntation and Masurnt, 59 (), [6 Shn, T.A., Tu, Y.Q., Zhang, H.T. (). A novl ti varying signal procssing thod for Coriolis ass flowtr. viw of Scintific Instrunts, 85 (6), 656. [7 Knapp, C.H., Cartr, G.C. (976). Th gnralizd corrlation thod fo stiation of ti dlay. IEEE Transactions on Acoustics, Spch and Signal Procssing, (), -7. [8 Cartr, G.C. (99). Cohrnc and Ti Dlay Estiation: An Applid Tutorial for sarch, Dvlopnt, Tst, and Evaluation Enginrs. IEEE Prss. [9 Maskll, D.L., Woods, G.S. (5). Adaptiv subsapl dlay stiation using a odifid quadratur phas dtctor. IEEE Transactions on Circuits and Systs, 5 (), [ So, H.C. (). Ti-dlay stiation for sinusoidal signals. IEE Procdings - adar, Sonar and avigation, 8 (6), 8-. [ Maskll, D.L., Woods, G.S. (). Th discrt-ti quadratur subsapl stiation of dlay. IEEE Transactions on Instruntation and Masurnt, 5 (), -7. [ So, H.C. (5). A coparativ study of two discrt-ti phas dlay stiators. IEEE Transactions on Instruntation and Masurnt, 5 (6), 5-5. [ Huang, C., Jiang, Y.Q. (5). Iprovd window and intrpolation algorith for analysis of powr syst haronics. Procdings of th CSEE, 5 (5), 6-. [ Oppnhi, A.V., Schafr,.W. (989). Discrt-Ti Signal Procssing. Prntic-Hall. civd May 9,. Accptd Octobr,. 56