Design of Variable Fractional-Delay FIR Differentiators

Transcription

1 Worl Acy of Scic, Egirig chology Dsig of Vril Frctiol-Dly FIR Diffrtitors Jog-Jy Shyu, Soo-Chg Pi i-h Chg Astrct I this r, th lst-squrs sig of vril frctiol-ly (VFD) fiit iuls rsos (FIR) igitl iffrtitors is roos. h us trsfr fuctio is forult so tht Frrow structur c li to rliz th sig syst. Also, th sytric chrctristics of filtr cofficits r riv, which ls to th colxity ructio y svig lost hlf of th ur of cofficits. orovr, ll th lts of rlt vctors or trics for th otil rocss c rrst i clos fors, which th sig sir. Dsig xl is lso rst to illustrt th ffctivss of th roos tho. ywors Diffrtitor, vril frctiol-ly filtr, FIR filtr, lst-squrs tho, Frrow structur. I. IRODUCIO IGIAL iffrtitors r grlly us to clcult th Disttous rt of chg of igitl sigls, hv wi lictios i istrut surt, ioicl girig sigl rocssig [1] [4] tc. For th st c, svrl tchiqus hv roos to sig igitl iffrtitors, icluig igfiltr roch, qurtic otiiztio, xflt tchiqu, Siso itgrtio rul, wto-cots forul, ilir trsfortio si-ifiit rogrig roch [5] [15] tc. Also, thr is rch of tr cocrig th sig of vril frctiol-ly (VFD) igitl filtrs which r li to th ly chrctristic to justl oli without sigig w filtrs [16] [4]. I this r, th sig of VFD FIR iffrtitors is roos. First, th us trsfr fuctio is forult such tht th iffrtitors c ilt y Frrow structur [16]. h th sytry o filtr cofficits is lyz, it is fou tht ch of th sufiltrs i Frrow structur c sig y th covtiol lir-hs FIR filtrs with sytric/tisytric cofficits. I this r, lst-squrs tho is us to th otil sig of VFD FIR iffrtitors, ll of th lts of rlt vctors or trics c rrst i clos fors. Filly, sig xl is rst to ostrt th fficicy of th Jog-Jy Shyu is with Drtt of Elctricl Egirig, tiol Uivrsity of ohsiug, ohsiug, iw, ROC (-il: jshyu@u.u. tw). Soo-Chg Pi is with Drtt of Elctricl Egirig, tiol iw Uivrsity, ii, iw, ROC (-il: i@cc..tu.u.tw ). i-h Chg is with Drtt of Elctricl Egirig, tiol Uivrsity of ohsiug, ohsiug, iw, ROC (-il: @ il. u. u.tw). roos tho. Also, it is fou tht lthough th xistig itrtiv wight-lst-squrs tho such s th tchiqu i [5] c li to iiiz th solut rror of vril frqucy rsos, ut it is ot rco cus th ly rsoss r vry ssitiv to th vritio of gitu rsoss rou.c. frqucy. II. PROBLE FORULAIO AD DESIG EXAPLE Covtiolly, for sigig iffrtitor th sir frqucy rsos is grlly giv y H c ji j, (1) I is th rscri ly is th cutoff frqucy. I this r, it is xt to sig VFD FIR iffrtitor, th sir vril frqucy rsos cos ji H, j,, () is justl rtr which is us to cotrol th ly of iffrtitor oli. I this r, th us trsfr fuctio c chrctriz y, () 0 H z h z th cofficits h r xrss s th olyoils of h h,, (4) 0 hc () c forult ito,, 0 0 G z 0 G H z h z 5 th sufiltrs z r giv y G z h, z, 0 (6) 0 th trsfr fuctio c ilt y Frrow structur [16]. otic tht () c furthr rrst y 716

2 Worl Acy of Scic, Egirig chology ji, si cos H j ji j 0 j! 1 j ji 1 (7) 0! for sufficit lrg. Corig (5) (7), it c fou tht th frqucy rsos of th sufiltr G z is ihrtly us to roxit 1 ji 1 j!, for 0. So it is rsol to choos th iuls rsos of G z to tisytric for v sytric for o, th frqucy rsos of (5) for v c forult ito H j, j 1 o j,, 1 h h / o / j 1, si, cos j o 1, for v, o, for o, h, h,,, (8) (9) 1, 0 (10) h,1, 0, 0, o (10) h,1 h,1, 1, 0. o It is ot tht h, 0 for 0, oviously I. Also, th cs for o c xt i th s r. Dfi 1,0,,,0,, 1,,,,, (11) 0,0,,,0,, 0,,,,, o o, si,,si,, si,, si (11) s (11c) 1 1 o o c,,, cos,,,, cos (11) th surscrit ots th trsos ortor, Eq.(8) c furthr rrst y j j H, j s, + c,. (1) I this r, th ojctiv rror fuctio is giv y 0.5 j,,, H H j cos si, c, j s cos, (1) s (14) 0.5 si c,. (14) Eq.(14) c furthr rrst y s r Q (15) 0.5 s cos, (16) r cos s, (16) Q s, s,. (16c) By th ylor sris xsio of cosi fuctio, cos s ! ! 1 th lts of r r 0.5 cos si si 1 i Q c riv s follows: i si, 0 0!! 1 (17) 0 1 1, (17) 717

3 Worl Acy of Scic, Egirig chology Q 0.5 ˆ, si siˆ i l ˆ si ˆ ˆ si, ˆ 1 ˆ ˆ 0 il, 1 1. (17c) I(17), i ˆ l / i o, 1, o,, l / ˆ o, 1 x y ots th rir wh th itgr x is ivi y th itgr y, u ots th lrgst itgr lss th or qul to th rl ur u. Also, th i (17) ust chos lrg ough s i [], 10 is us i this r. Siilrly, s r Q (18) () s 6! 1 (19) () th lts of r r i o Q r giv y 4 cos, i 1! , (19) Q o ˆ ˆ ˆ 0.5 si si il,, ˆ ˆ ˆ 0 il, (19c) (c) i, 1 I (19), o i, 1 l ˆ. 1, ˆ o l, 1 Oc (17) (19) r fou, th cofficit vctors c tri y iffrtitig (1) with rsct to,, r Q,, r Q th sttig th ov rsults to zros, which yils (0) (0) 1-1 Q r (1) 1-1 Q r. (1) o vlut th rforc of th sig tho, th () Fig. 1 Dsig of 50, 7, 0.9 VFD FIR iffrtitor. () orliz root--squrs rror of th vril frqucy rsos, th xiu solut rror of vril frqucy rsos th xiu solut ly rror r fi y 0.5 j H, H, %, 0.5 H, gitu rsoss. () Dly rsoss. (c) Asolut rrors of vril frqucy rsoss. () Asolut ly rrors. 1/ () 718

4 Worl Acy of Scic, Egirig chology j x H, H,, 0, () x,,, 0, , (c) rsctivly,,, r th il ly rsos th ctul ly rsos of th sig VFD FIR iffrtitor. o cout () (c), th frqucy th rtr r uiforly sl t th st sizs /400 1/50, rsctivly. otic tht th ly rrors for 0, r ot iclu i (c) cus th gitu rsoss ovr tht r r lost zro. Exl: his xl ls with th sig of 50, 7, 0.9 VFD FIR iffrtitor, th gitu rsoss ly rsoss r show i Fig.1() (), rsctivly, whil Fig.1(c) () rst th solut rrors of vril frqucy rsoss th solut ly rrors, rsctivly. h rlt rrors r list s low: % For th VFD FIR iffrtitor sig, it is ot rco to icorort wightig fuctio i (1) such tht th solut rror of vril frqucy rsoss c iiiz, cus th ly rsoss r 0 r vry ssitiv to th vritio of gitu rsoss. For xl, if th itrtiv wight-lst-squrs tho i [5] is li, out fiv itrtios r to iiiz th xiu solut rror of vril frqucy rsoss, which is uch sllr th tht show ov, ut th xiu solut ly rror cos which is uch lrgr th tht show ov. III. COCLUSIOS I this r, th roos lst-squrs tho hs succssfully li to th sig of VFD FIR iffrtitors. It c fou tht th oit iffrc tw grl VFD FIR igitl filtrs [] VFD FIR iffrtitors is tht oly th il cofficit of th first sufiltr is rquir for th forr ut ot for th lttr. Although th roos syst c rlc y cscig iffrtitor with vril frctiol-ly filtr, ut howvr, which will ls to lrgr ly. Oviously, th tchiqu roos i this r c lso li to th sig of VFD highr-orr or frctiol-orr FIR iffrtitors. REFERECES [1] S. Usui I. Airor, Digitl low-ss iffrtitio for iologicl sigl rocssig, IEEE rs. Bio. Eg., vol.be-9, o.10, , Oct [] R. C. vgh, FIR iffrtitors for qutiz sigls, IEEE rs. Sigl Procss., vol.49, o.11,.71-70, ov [] J. Luo, J. Bi, P. H. Yig, Axil stri clcultio usig low-ss igitl iffrtitor i ultrsou lstogrhy, IEEE rs. Ultrso., Frrolct., Frq. Cotr., vol.51, o.9, , S [4] S. L. Li S. our, O-chi ris-ti surt, IEEE rs. Istru. s., vol.5, o.6, , Dc [5] S.-C. Pi J.-J. Shyu, Eigfiltr sig of high-orr igitl iffrtitors, IEEE rs. Acoust., Sch, Sigl Procss., vol.7, o.4, , Ar [6] S. Sur, Y. Su, A. Atoiou W.-S. Lu, Dsig of igitl iffrtitors stisfyig rscri scifictios usig otiiztio tchiqus, i Proc. ISCAS 1989, y 1989, [7] B. ur, S. C. D. Roy H. Shh, O th sig of FIR igitl iffrtitors which r xilly lir t th frqucy /, ositiv itgrs, IEEE rs. Sigl Procss., vol.40, o.9,.4-8, S [8] E. Hrowicz. Rojwsi, Dsig of FIR first orr igitl iffrtitors of vril frctiol sl ly usig xilly flt rror critrio, Elctro. Ltt., vol.0, o.1,.17-18, J [9]. A. Al-Aloui, ovl IIR iffrtitor fro th Sio itgrtio rul, IEEE rs. Circuits Syst. I, Fut. hory Al., vol.41, o., , F [10] X. Zhg. Yoshiw, Dsig of full IIR igitl iffrtitors, i Proc. IEEE It. Cof. url t. Sigl Procss., Dc. 00, [11] C.-C. sg, Digitl iffrtitor sig usig frctiol ly filtr liit couttio, IEEE rs. Circuits Syst. I, Rg. Prs, vol.5, o.10,.48-59, Oct [1]. Q. go, A w roch for th sig of wi igitl itgrtor iffrtitor, IEEE rs. Circuits Syst. II, Ex. Brifs, vol.5, o.9, , S [1]. A. Al-Aloui, Lir hs low-ss IIR igitl iffrtitors, IEEE rs. Sigl Procss., vol.55, o., , F [14] C.-C. sg S.-L. L, Dsig of igitl iffrtitor usig iffrc forul Richrso xtroltio, vol., o., , Ju [15] C. Y.-F. Ho, B. W.-. Lig, Y.-Q. Liu, P..-S..-L. o, Otiu sig of iscrt-ti iffrtitors vi si-ifiit rogrig roch, IEEE rs. Istru. s., vol.57, o.10,.6-0, Oct [16] C. W. Frrow, A cotiuously vril igitl ly lt, i Proc. ISCAS 1988, y 1988, [17]. I. Lso, V. Väliäi,. rjli U.. Li, Slittig th uit ly: ools for frctiol ly filtr sig, IEEE Sigl Procss. g., vol.1,.0-60, J [18] W.-S. Lu.-B. Dg, A irov wight lst-squrs sig for vril frctiol ly FIR filtrs, IEEE rs. Circuits Syst. I, Alog Digit. Sigl Procss., vol.46, o.8, , Aug [19] H. Johsso P. Löworg, O th sig of justl frctiol ly FIR filtrs, IEEE rs. Circuits Syst. II, Alog Digit. Sigl Procss., vol.50, o.4, , Ar. 00. [0] C.-C. sg, Dsig of vril frctiol ly FIR filtr usig sytry, i Proc. ISCAS 004, y 004,.III/477-III/480. [1] H. Zho J. Yu, A sil fficit sig of vril frctiol ly FIR filtrs, IEEE rs. Circuits Syst. II, Ex. Brifs, vol.5, o., , F [].-B. Dg, Y. Li, Wight-lst-squrs sig of vril frctiol-ly FIR filtrs usig cofficit sytry, IEEE rs. Sigl Procss., vol.54, o.8,.0-08, Aug [] E. Hrowic H. Johsso, A colx vril frctiol-ly FIR filtr structur, IEEE rs. Circuits Syst. II, Ex. Brifs, vol.54, o.9, , S [4].-B. Dg, Sytric structurs for o-orr xilly flt wight-lst- squrs vril frctiol-ly filtrs, IEEE rs. Circuits Syst. I, Rg. Prs, vol.54, o.1,.718-7, Dc. 007 [5] C. Y. Chi Y.. ou, A w slf-iitit otiu WLS roxitio tho for th sig of lir hs FIR igitl filtrs, i Proc. ISCAS 1991, Ju 1991,

5 Worl Acy of Scic, Egirig chology Jog-Jy Shyu ws or i iw, o rch 7, H rciv th B.S. gr i lctricl girig fro tug Uivrsity, ii, iw i 198 th.s. Ph.D. grs i lctricl girig fro tiol iw Uivrsity, ii i , rsctivly. H ws with th Drtt of Coutr Scic Egirig, tug Uivrsity s Associt Profssor i 199 Profssor i Fro 1997 to 000, h ws with th Drtt of Coutr Couictio Egirig, tiol ohsiug First Uivrsity of Scic chology, ohsiug, iw. H is currtly with th Drtt of Elctricl Egirig, tiol Uivrsity of ohsiug, ohsiug. His rsrch itrsts iclu th sig ilttio of igitl filtrs, igitl sigl rocssig. Soo-Chg Pi ws or i Soo-Auo, iw, i H rciv th B.S.E.E. gr fro tiol iw Uivrsity, ii, iw, i 1970 th.s.e.e. Ph.D. grs fro th Uivrsity of Clifori, St Brr, i , rsctivly. H ws Egirig Officr with th Chis vy Shiyr fro 1970 to Fro 1971 to 1975, h ws Rsrch Assistt with th Uivrsity of Clifori. H ws Profssor with th Chir of th Drtt of Elctricl Egirig, tug Istitut of chology, tiol iw Uivrsity, fro 1981 to to 1998, rsctivly. H is currtly th D of th Collg of Elctricl Egirig Coutr Scic Profssor of th Drtt of Elctricl Egirig th Grut Istitut of Couictio Egirig, tiol iw Uivrsity. His rsrch itrsts iclu igitl sigl rocssig, ig rocssig, oticl ifortio rocssig, lsr hologrhy. Dr. Pi rciv th tiol Su Yt-S Acic Achivt Awr i Egirig i 1984, th Distiguish Rsrch Awr fro th tiol Scic Coucil fro 1990 to 1998, th Outstig Elctricl Egirig Profssor Awr fro th Chis Istitut of Elctricl Egirig i 1998; th Acic Achivt Awr i Egirig fro th iistry of Euctio i 1998, th P W-Yu Distiguish Rsrch Awr i 00, th tiol Chir Profssor Awr fro th iistry of Euctio i 00. H ws th Prsit of th Chis Ig Procssig Pttr Rcogitio Socity i iw fro 1996 to1998 is r of th IEEE, Et u th Oticl Socity of Aric. H c IEEE Fllow i 000 for his cotriutios to th vlot of igitl igfiltr sig, color ig coig, sigl corssio to lctricl girig uctio i iw. i-h Chg rciv th B.S gr i Elctricl Egirig fro tiol Uivrsity of ohsiug, iw, i 007, h is currtly worig towr th.s. gr. His rsrch itrsts iclu filtr sig igitl sigl rocssig. 70