Chapter 5. Chapter 5 125

Transcription

1 Chptr 5 Chptr 5: Itroductio to Digitl Filtrs Itroductio No rcursiv digitl filtrs FIR Rcursiv digitl filtr IIR Digitl Filtr Rlistio Prlll rlistio Cscd rlistio Mgitud d Phs Rspos Miimum-phs, Mximum-phs d Mixd phs systms All-Pss Filtrs A scod Ordr Rsot Filtr Stility of scod-ordr filtr Digitl Oscilltors Si d cosi oscilltors Notch filtrs Summry... 8 Chptr 5: Prolm Sht 5 Chptr 5 5

2 Chptr 5: Itroductio to Digitl Filtrs 5. Itroductio Thr r two typs of digitl filtrs:. Rcursiv thr is t lst o fdc pth i th filtr. No-rcursiv o fdc pths A lir tim ivrit LTI discrt systm dscrid y th followig qutio is commoly clld digitl filtr: y M x y Fd forwrd L whr x[] is th iput sigl, y[] is th output sigl.,,,..., M ;,,,..., L r costts filtr cofficits. Ths cofficits dtrmi th chrctristics of th systm. Fdc 5. wh = wh th filtr is sid to o-rcursiv typ rcursiv typ. Chptr 5 6

3 Chptr No rcursiv digitl filtrs FIR If =, th th clcultio of y[] dos ot rquir th us of prviously clcultd smpls of th output s qutio 5.. x x x x y M M This is rcogisd s covolutio sum. M x h y Thrfor th impuls rspos is idticl to th cofficits, tht is, othrwis M h x x x x y M M Ay filtr tht hs impuls rspos of fiit durtio is clld Fiit Impuls Rspos FIR filtr. Exmpl: X Y x x x y This is o-rcursiv scod ordr FIR filtr h h hm

4 Proprty: A proprty of th FIR filtr is tht it will lwys stl. Stility rquirs tht thr should o pols outsid th uit circl. This coditio is utomticlly stisfid sic thr r o pols t ll outsid th origi. I fct, ll pols r loctd t th origi. Aothr proprty of o-rcursiv filtr is tht w c m filtrs with xctly lir phs chrctristics Not: Th ility to hv xctly lir phs rspos is o of th most importt proprtis of LTD systm filtr. Wh sigl psss through filtr, it is modifid i mplitud d/or phs. Th tur d xtt of th modifictio of th sigl is dpdt o th mplitud d phs chrctristics of th filtr. Th phs dly or group dly of th filtr provids usful msur of how th filtr modifid th phs chrctristic of th sigl. If w cosidr sigl tht cosists of svrl frqucy compots g. spch wvform th phs dly of th filtr is th mout of tim dly ch frqucy compot of th sigl suffrs i goig through th filtr. phs _ dly T p 5. [th gtiv of th phs gl dividd y frqucy] Chptr 5 8

5 Th group dly o th othr hd is th vrg tim dly th composit sigl suffrs t ch frqucy s it psss from th iput to th output of th filtr. group _ dly T g d d 5. [th gtiv of th drivtiv of th phs with rspct to frqucy] = - - Figur 5.: Phs rspos of lir phs filtr A costt group dly ms tht sigl compots t diffrt frqucis rciv th sm dly i th filtr. Chptr 5 9

6 A lir phs filtr givs sm tim dly to ll frqucy compots of th iput sigl. A filtr with olir phs chrctristic will cus phs distortio i th sigl tht psss through it. This is cus th frqucy compots i th sigl will ch dlyd y mout ot proportiol to frqucy, thry ltrig thir hrmoic rltioship. Such distortio is udsirl i my pplictios, for xmpl music, vido tc. A filtr is sid to hv lir phs rspos if its phs rspos stisfis o of th followig rltioships: 5. whr d r costts. Exmpl: Two filtr structurs r show low. Show tht oth filtrs hv lir phs. x[] - - x[] y[] y[] Chptr 5

7 Chptr Pol-ro Pttrs of Lir Phs Filtrs Lir phs filtrs provid costt dly with o mplitud distortio. A FIR filtr with its impuls rspos symmtric out th midpoit is dowd with lir phs d costt dly. For lir phs FIR filtr, th pols must li t th origi = if th squc h[] is to of fiit lgth. Squcs tht r symmtric out th origi i.. = rquir ] [ ] [ h h d thus phs: = - lir phs phs: = / - lir phs cos x x x y si si x x y

8 Th ros of lir-phs squc must occur i rciprocl pirs d xhiit cougt rciprocl symmtry s show low. = = r /r r r For rl ro A ro o th rl xis is pird with ust its rciprocl r /r /r Complx cougts For complx ro Ech ro ot o th rl xis is pird with its rciprocl d its cougt. Zros t = or t =- c occur sigly, cus thy form thir ow rciprocl d thir ow cougt. If thr r o ros t =, lir-phs squc is lwys v symmtric out its midpoit. For odd symmtry out th mid-poit, thr must odd umr of ros t =. Th frqucy rspos of lir phs filtr my writt s A for v symmtry or A A for odd symmtry. Chptr 5

9 Exmpl: Is this lir phs filtr? Stch th pol-ro plot. h,,, = = = = All of its pols r t =; Its ros r t =- d =-.5±.866 Th rl ro t =- c occur sigly Complx cougt pir of ros li o th uit circl => Th impuls rspos h[] cot symmtric out th origi, v though it is symmtric out its midpoit =.5 Im -.5,.866 = - R -.5, Lir phs squc with v symmtry out its mid poit. Chptr 5

10 Exmpl:.5 h[ ] {,,.5,,} =- =- = = = Is lir phs? Stch th pol-ro plot of. Sic h[] is v symmtric out = with h[]=h[-], w hv =/ Im.5 = -.5 R Chptr 5

11 5... Typs of Lir-phs Squcs Lir phs squcs fll ito typs: Typ : Squc hs v symmtry d odd lgth. Typ squc must hv v umr of ros t =- if prst d = if prst. = Im - - R Ev umr of ros Pol-ro plot All othr ros must show cougt rciprocl symmtry. Typ : Squc hs v symmtry d v lgth. Typ squc must hv odd umr of ros t =- if prst d v umr of ros t = if prst. Im Cougt rciprocl symmtry = Ev umr of ros Odd umr of ros - R Chptr 5 5

12 Typ : Squc hs odd symmtry d odd lgth. Typ squc must hv odd umr of ros t =- d odd umr of ros t =. Im = - Odd umr of ros R Cougt rciprocl symmtry Typ : Squc hs odd symmtry d v lgth. Typ squc must hv odd umr of ros t =. Th umr of othr ros, if prst t =-, must v. Im = Odd umr of ros Ev umr of ros - R Chptr 5 6

13 Exmpl: Fid ll of th ro loctios of typ lir-phs squc if it is ow tht thr is ro t d ro t. Im = /r= r=/ - r=/ 6º R Ev umr of ros /r= Pol-ro plot Du to cougt rciprocl symmtry, th ro t implis w hv ro t, d. For typ squc, th umr of ros t =- must v, so thr must othr ro t =-. Thus, thr r 6 ros. Chptr 5 7

14 Chptr Rcursiv digitl filtr IIR Evry rcursiv digitl filtr must coti t lst o closd loop. Ech closd loop cotis t lst o dly lmt. L M y x y For rcursiv digitl filtrs. Lt =, = for > d = & = for >. y x y IIR filtr A rcursiv filtr is ifiit impuls rspos filtr IIR. Exmpl: d ordr FIR filtr d ordr IIR filtr ll pol filtr IIR filtr Zros oly Pols d Zros oly

15 Not: Pols d ros c rl or imgiry Exmpl: Th diffrc qutio is: y[] = x[] y[-]. Th DC gi of c otid y sustitutig =. If dc gi is udsirl, itroduc costt gi fctor of -, so tht coms dc gi = y[] = -x[] y[-] Exmpl: Cosidr lowpss filtr y[] = y[-] x[], < < i Dtrmi so tht =. ii Dtrmi th db dwidth hr for th ormlisd filtr i prt i. i Y = Y - X w hv = Chptr 5 9

16 Chptr 5 = - cos si cos si cos Scod Mthod: * cos cos c θ θ θ hlf-powr poit c c cos cos c c db db

17 Chptr 5 Exmpl: Cosidr filtr dscrid y c c whr & c r costts. Show tht th mgitud rspos is uity for ll. ] [ ] [ * c c c c c c c c This is ll-pss filtr. -

18 Chptr 5 5. Digitl Filtr Rlistio structur ros M pols L structur pols L ros M L M L M L M X Y Y X Y y x y

19 X Y M - L - Figur 5.: Structur or Dirct Form X Y L - - M Figur 5.: Structur or Dirct Form II Chptr 5

20 I th cs wh L = M, w hv coic form rlistio. X Y L - M Figur 5.: Coic form A discrt-tim filtr is sid to coic if it cotis th miimum umrs of dly lmts cssry to rlis th ssocitd frqucy rspos. 5.. Prlll rlistio i i prlll_ structur us prtil frctio to oti i M M L L... Chptr 5

21 X Y Figur 5.5: Prlll structur 5.. Cscd rlistio M M L L ˆ i ˆ ˆ ˆ ˆ... i cscd_ structur Product of lowr ordr trsfr fuctio i. st or d ordr sctios Th cscd structur is th most populr form X ˆ ˆ ˆ Y Figur 5.6: Cscd structur Chptr 5 5

22 Chptr 5 6 Exmpl: A prlll rlistio of third ordr systm is giv y D C B A

23 Chptr 5 7 Exmpl: A cscd rlistio of third-ordr systm is giv y x[] y[]

24 x[].5.6 y[] Cscd Exmpl: Implmt th followig systm i th cscd, dirct form II d prlll structurs. All cofficits r rl.. x[] y[] cscd structur x[] y[] -- - Dirct form II - Chptr 5 8

25 A B x[] y[] Prlll structur. x[] y[] cscd structur No prlll structur xists cus prtil frctio xpsio cot prformd. Chptr 5 9

26 x[] y[] Dirct Form II - c. prlll cscd x[] - - y[] - - prlll structur Chptr 5 5

27 Chptr Mgitud d Phs Rspos W c show tht th mgitud rspos is v fuctio of frqucy. Th phs rspos is odd fuctio of frqucy. Exmpl: Clcult th mgitud d phs rspos of th - smpl vrgr giv y othrwis h h h - y[] x[] -

28 cos Prcutios must t wh dtrmiig th phs rspos of filtr hvig rl-vlud trsfr fuctio, cus gtiv rl vlus produc dditiol phs of rdis. For xmpl, lt us cosidr th followig lir-phs form of th trsfr fuctio = - B rl-vlud fuctio of tht c t positiv d gtiv vlus. B cos B si B cos B si Lt phs gl : B si t t B cos t = t- = - or = - phs gl Th phs fuctio icluds lir phs trm d lso ccommodts for th sig chgs i B. Sic - c xprssd s, phs umps of will occur t frqucis whr B chgs sig. If B >, th = -. If B <, th = -.. Chptr 5 5

29 Lt us gt c to our xmpl [ cos ] [ cos ] d Th pproprit sig of must chos to m odd fuctio of frqucy. Ev fuctio - - -/ / Odd fuctio - - -/ / - Exmpl: Fid th mgitud d phs rspos of th followig: h, h, h, h, othrwis. Chptr 5 5

30 Chptr 5 5 ] cos [ B Exmpl: cs othrwis Th mplitud fuctio is vr gtiv thrfor thr is o phs umps of Ev fuctio - - Odd fuctio

31 h[] = [-] = - = B = = - Not: Wh phs xcds rg ump of is dd to rig th phs c ito rg. Phs Jumps: From th prvious xmpls, w ot tht thr r two occsios for which th phs fuctio xprics discotiuitis or umps.. A ump of occurs to miti th phs fuctio withi th pricipl vlu rg of [- d ]. A ump of occurs wh B udrgos chg of sig Th sig of th phs ump is chos such tht th rsultig phs fuctio is odd d, ftr th ump, lis i th rg [- d ]. Chptr 5 55

32 Chptr 5 56 Exmpl: Mgitud d phs rspos of cusl -smpl vrg. ] cos [ ; ] cos [ othrwis for B B B h B Phs is udfid t poits =. Exmpl: Dtrmi d stch th mgitud d phs rspos of th followig filtrs: i x x y ii 8 x x y iii x y / / / -/

33 Chptr 5 57 i si si ] [ ] [ ] [ X X Y - / -/ -

34 Chptr 5 58 ii si si B X Y X X Y / / / -/ / / / /

35 Chptr 5 59 iii X Y x y Exmpl: Dtrmi d stch th mgitud d phs rspos of st ordr rcursiv filtr IIR filtr y x y phs X Y cos si t cos si cos cos cos si t cos si cos cos - - = - /

36 Mgitud: = * [ * is th complx cougt] Assumig < < cos Ev Symmtry =.5 - Odd Symmtry - No-lir phs Exmpl: Low pss filtr Th gi c slctd s, so tht th filtr hs uity gi t =. Chptr 5 6

37 I this cs, for uity gi t =. Th dditio of ro t = - furthr ttuts th rspos of th filtr t high frqucis Lowpss filtr - c W c oti simpl highpss filtrs y rflctig foldig th pol-ro loctios of th lowpss filtrs out th imgiry xis i th -pl. igh pss filtr - d y x x lowpss filtr Chptr 5 6

38 5 y x x ighpss filtr 5 5 = -cos - f = -cos - g = 8-cos - Chptr 5 6

39 5. Miimum-phs, Mximum-phs d Mixd phs systms Lt us cosidr two FIR filtrs: = -.5 ρ = = is th rvrs of th systm. This is du to th rciprocl rltioship tw th ros of d. & 5 cos Th mgitud chrctristics for th two filtrs r idticl cus th roots of d r rciprocl. Phs: φ θ t φ θ t siθ cosθ siθ cosθ Chptr 5 6

40 Not: If w rflct ro with mgitud = tht is isid th uit circl ito ro with mgitud outsid th uit circl th mgitud chrctristic of th systm is ultrd, ut th phs rspos chgs. W osrv tht th phs chrctrs gis t ro phs t frqucy = d trmits t ro phs t th frqucy =. c th t phs chg. Miimum phs filtr O th othr hd, th phs chrctristic for th filtr with th ro outsid th uit circl udrgos t phs chg rdis As cosquc of ths diffrt phs chrctristics, w cll th first filtr miimum-phs systm d th scod systm is clld mximum-phs systm. If filtr with M ros hs som of its ros isid th uit circl d th rmiig outsid th uit circl, it is clld mixd-phs systm. A miimum-phs proprty of FIR filtr crris ovr to IIR filtr. Lt us cosidr B A is clld miimum phs if ll its pols d ros r isid th uit circl. = R Miimum phs Chptr 5 6

41 If ll th ros li outsid th uit circl, th systm is clld mximum phs. = R Mximum phs If ros li oth isid d outsid th uit circl, th systm is clld mixd-phs. R Mixd phs = Not: For giv mgitud rspos, th miimum-phs systm is th cusl systm tht hs th smllst mgitud phs t vry frqucy. Tht is, i th st of cusl d stl filtrs hvig th sm mgitud rspos, th miimum-phs rspos xhiits th smllst dvitio from ro phs. Exmpl: Cosidr fourth-ordr ll-ro filtr cotiig doul complx cougt st of ros loctd t.7. Th miimum-phs, mixd phs d mximum phs systm pol-ro pttrs hvig idticl mgitud rspos r show low. Chptr 5 65

42 = = = =.7 / Miimum-phs mixd-phs mximum-phs Th mgitud rspos d th phs rspos of th thr systms r show low: Th miimum-phs systm sms to hv th phs with th smllst dvitio from ro t ch frqucy. miimum phs mixd-phs I th cs lir phs - mximum phs Chptr 5 66

43 Exmpl: A third ordr FIR filtr hs trsfr fuctio G giv y G 6 5 From G, dtrmi th trsfr fuctio of FIR filtr whos mgitud rspos is idticl to tht of G d hs miimum phs rspos. G 5 G 5 > Im 5 5 = R Th miimum phsfiltr P 5 Chptr 5 67

44 Chptr All-Pss Filtrs A ll-pss filtr is o whos mgitud rspos is costt for ll frqucis, ut whos phs rspos is ot idticlly ro. [Th simplst xmpl of ll-pss filtr is pur dly systm with systm fuctio = - ] A mor itrstig ll-pss filtr is o tht is dscrid y L L L L L L, whr = d ll cofficits r rl. If w dfi th polyomil A s L L A A A i.. ll pss filtr. Furthrmor, if is th modulus of pol of, th / is th modulus of ro of {i.. th modulus of pols d ros r rciprocls of o othr}. Th figur show low illustrts typicl pol-ro pttrs for sigl-pol, siglro filtr d two-pol, two-ro filtr. = All-pss filtr = r All pss filtr /r, /r, - r, -

45 Chptr 5 69 < for stility W c sily show tht th mgitud rspos is costt. * cos cos Phs rspos: cos si t cos si cos Wh < <, th ro lis o th positiv rl xis. Th phs ovr is positiv, t = it is qul to d dcrss util =, whr it is ro. Wh -< <, th ro lis o th gtiv rl xis. Th phs ovr is gtiv, strtig t for = d dcrss to - t =. - =.5 = -.5 = -.8

46 5.6 A scod Ordr Rsot Filtr x[] y[] r p - - p - - p p r r r cos r cos r si r si A All pol systms hs pols oly without coutig th ros s th origi r r p p r r cos r r B Comprig A d B, w oti r cos r Cos f s f = rsot frqucy Chptr 5 7

47 5.7 Stility of scod-ordr filtr Cosidr two-pol rsot filtr giv y & r cofficits This systm hs two ros t th origi d pols t p, p Th filtr is stl if th pols lis isid th uit circl i.. p < & p < For stility <. If = th th systm is oscilltor Mrgilly stl Assum tht th pols r complx i. < < d If th w gt rl roots. Th stility coditios dfi rgio i th cofficit pl, which is i th form of trigl s low Th systm is oly stl if d oly if th poit, lis isid th stility trigl. Chptr 5 7

48 Chptr 5 7 Stility Trigl If th two pols r rl th thy must hv vlu tw - d for th systm to stl. Th rgio low th prol > corrspods to rl d distict pols. Th poits o th prol = rsult i rl d qul doul pols. Th poits ov th prol corrspod to complxcougt pols = Rl Pols Complx Cougt Pols prol = d d d

49 5.8 Digitl Oscilltors A digitl oscilltor c md usig scod ordr discrttim systm, y usig pproprit cofficits. A diffrc qutio for oscilltig systm is giv y p Acos From th tl of -trsforms w ow tht th -trsform of p[] ov is P cos cos P Y X cos cos Lt, Tig ivrs -trsform o oth sids, w oti y cos y y x cosx No Iput trm for oscilltor x[] =, x[-] = So th qutio of th digitl oscilltor coms cos y y y Chptr 5 7

50 d its structur is show low. y[] = A cos y[-] - - y[-] = cos = - To oti y[] = A cos, us th followig iitil coditios: y[] = A cos. = A y[-] = A cos-. = A cos Th frqucy c tud y chgig th cofficit is costt. Th rsot frqucy of th oscilltor is, cos For oscilltor = Exmpl: A digitl siusoidl oscilltor is show low. x[] - y[] = A si Assumig is th rsot frqucy of th digitl oscilltor, fid th vlus of d for sustiig th oscilltio. Chptr 5 7

51 K r cos θ K r r θ K r θ r θ K θ r = - r cos ; = r For oscilltio = r = = - cos Writ th diffrc qutio for th ov figur. Assumig x[] = Asi [], d y- = y- =. Show, y lysig th diffrc qutio, tht th pplictio of impuls t = srvs th purpos of giig th siusoidl oscilltio, d prov tht th oscilltio is slfsustiig thrftr. y y y y x cos y y Asi = y[] = cos y[-] y[-] A si [] y[] = A si = y[] = cos y[] y[-] A si [] y[] = cos Asi = A si = y[] = cos y[] y[] A si [] = cos A si A si = A cos [ si cos ] A si = A si [ cos ] = A[si si ] Chptr 5 75

52 whr si = si si y[] = A si d so forth. c By sttig th iput to ro d udr crti iitil coditios, siusoidl oscilltio c otid usig th structur show ov. Fid ths iitil coditios. y cos y y x x[] = for oscilltor = y[] = cos y[-] y[-] for oscilltio, y[-] = o cosi trms y[] = -y[-] y[-] = -Asi si trm is rquird y[] = --A si = A si Iitil coditios: y[-] = ; y[-] = -Asi Chptr 5 76

53 5.8. Si d cosi oscilltors Siusoidl oscilltors c usd to dlivr th crrir i modultors. I modultio schms, oth sis d cosis oscilltors r dd. A structur tht dlivrs sis d cosis simultously is show low: - si cos y[]= cos -si cos x[]= si - Proof: Trigoomtric qutio for cos is: cos = coscos - si si Lt y[] = cos d x[] = si y[] = cos y[] si x[] Rplc y - y[] = cos y[-] si x[-] A Similrly si = si cos si cos x[] = si y[] x[] cos Rplc - x[] = si y[-] x[-] cos B Usig qutios A & B ov, th structur show ov c otid. Chptr 5 77

54 Exrcis: A oscilltor is giv y th followig coupld diffrc qutios xprssd i mtrix form. y y c s cos si si y cos y c s Drw th structur for th rlistio of this oscilltor, whr is th oscilltio frqucy. If th iitil coditios y c [-] = Acos d y s [-] = -Asi, oti th outputs y c [] d y s [] usig th ov diffrc qutios. y y c s cos yc si ys si y cos y c s - cos y c [] si -si cos - = y s [] = si A cos cos -Asi = = y c [] = cos Acos - si -Asi = A = y c [] = cos.a - si. = Acos = y s [] = A si = A si = y c [] = cos y c [] - si y s [] = cos A cos - si A si = A cos = y c [] = A cos similrly y s [] = A si Chptr 5 78

55 Exrcis: For th structur show low, writ dow th pproprit diffrc qutios d hc stt th fuctio of this structur. - si y [] Notch filtrs Wh ro is plcd t giv poit o th -pl, th frqucy rspos will ro t th corrspodig poit. A pol o th othr hd producs p t th corrspodig frqucy poit. Pols tht r clos to th uit circl giv ris lrg ps, whr s ros clos to or o th uit circl producs troughs or miim. Thus, y strtgiclly plcig pols d ros o th - pl, w c oti smpl low pss or othr frqucy slctiv filtrs otch filtrs. Exmpl: Oti, y th pol-ro plcmt mthod, th trsfr fuctio of smpl digitl otch filtr s figur low tht mts th followig spcifictios: Notch Frqucy: 5 d width of th Notch: ±5 Smplig frqucy: 5 Chptr 5 79

56 Th rdius, r of th pols is dtrmid y: f r f s f 5 5 f To rct th compot t 5, plc pir of complx ros t poits o th uit circl corrspods to 5. i.. t gls of To chiv shrp otch filtr d improvd mplitud rspos o ithr sid of th otch frqucy, pir of complx cougt ros r plcd t rdius r <. r f.97 f s 5 = 6 6 Chptr 5 8

57 cos cos. 5. Summry At th d of this chptr, it is xpctd tht you should ow: Typs of digitl filtrs FIR/IIR d thir proprtis. Covrsio from FIR/IIR diffrc qutios to trsfr fuctios d c gi. FIR No Rcursiv, ll-ro Filtrs o Udrstdig of phs dly d group dly o Dfiitio of lir phs filtrs IIR Rcursiv, ll-pol or pol-ro Filtrs o Cscdd, prlll, d coic structurs Clcultio of db cut-off frqucy d db dwidth for simpl first-ordr FIR d IIR filtr B l to plot th mgitud rspos of simpl first-ordr FIR d IIR filtr. B l to distiguish tw lowpss d highpss filtrs sd o th diffrc qutios or trsfr fuctios for oth FIR d IIR Chptr 5 8

58 Giv FIR filtr diffrc qutio or trsfr fuctio, l to drw th mgitud d phs rsposs, d l to xpli th rltioship tw mgitud d phss rsposs. Th diffrcs tw miimum, mximum d mixd phs filtrs. Th similrity i mgitud rsposs wh filtr ros r rflctd out th uit circl. All-pss filtrs: l to show tht thir mgitud rspos is costt ut thir phs rspos is o-ro. B l to driv th trsfr fuctio for scod ordr rsotor filtr, d l to lys its stility proprtis usig th stility trigl d pol positios. B l to udrstd th rg th filtr cofficits c t i ordr to prsrv stility. Pricipls of stl, mrgilly stl d ustl filtrs d qutios for digitl oscilltors. B l to drw th structur of digitl oscilltor tht c simultously produc si d cosi oscilltios, d to iitilis it corrctly. Udrstdig d dsig of otch filtrs. Chptr 5 8