Module 5: IIR and FIR Filter Design Prof. Eliathamby Ambikairajah Dr. Tharmarajah Thiruvaran School of Electrical Engineering & Telecommunications

Transcription

1 Modul 5: IIR ad FIR Filtr Dsig Prof. Eliathamby Ambiairaah Dr. Tharmaraah Thiruvara School of Elctrical Egirig & Tlcommuicatios Th Uivrsity of w South Wals Australia

2 IIR filtrs Evry rcursiv digital filtr IIR must cotai at last o closd loop. Each closd loop cotais at last o dlay lmt y M a x L y For rcursiv digital filtrs IIR b For o rcursiv digital filtrs FIR b b Gral digital filtr ELEC97

3 3 Exampls b b a a a b b a a a d ordr FIR filtr ros oly d ordr IIR filtr all pol filtr d ordr IIR filtr with pols ad ros

4 Exampl: Fid th DC gai of th followig filtr: /- a - <a< Th DC gai of ca b obtaid by substitutig o Η. a ; a If th DC gai is udsirabl, you may itroduc a costat gai factor of -a, so that bcoms a dc gai db a 4

5 ELEC97 5 Exrcis Slct such that th filtr has uity gai at. b b a a a

6 Exampl: Cosidr a lowpass filtr y a y- b x Dtrmi b so that. ot: < a < Dtrmi th 3 db badwidth c for th ormalisd filtr i part abov. a b a b a b a b ; ; [ ] a a a a a a po powr alf a a a a a c c c 4 cos cos it cos c / Lowpass filtr

7 Exrcis A first ordr digital filtr is dscribd by a b b a Assum a b < a, b < Aswr : Dtrmi, so that th maximum valu of Comput th 3-db badwidth of th filtr c a b cos ELEC97

8 Exampls: Lowpass filtr : a a a Lowpass filtr : a - Th additio of a ro at - furthr attuats th rspos of th filtr at high frqucis ighpass filtr : 3 a a W ca obtai simpl highpass filtr by rflctig th pol-ro locatios of th lowpass filtr about th imagiary axis. -

9 ELEC97 9 Exampls : 5 x x y filtr ighpass : 4 x x y filtr Lowpass cos 5

10 Digital Filtr Dsig Ovrviw Th most commo digital filtrs fall ito four mai catgoris Low Pass igh Pass Bad Pass Bad Stop FIR filtrs ca b dirctly dsigd i th trasform domai IIR filtrs ar most commoly dsigd by slctig a suitabl aalogu filtr trasfr fuctio, s, ad th trasformig it ito a quivalt digital filtr trasfr fuctio,, usig a mappig fuctio

11 Idal Filtr Magitud Rsposs c c Low Pass Filtr c c igh Pass Filtr c c c c Bad Pass Filtr Bad Stop Filtr ELEC97

12 Spcificatio of a Digital filtr Th maority of filtr dsig tchiqus ar basd o th dsigr providig a spcificatio for a dsird frqucy Magitud Rspos Eg. For a low pass filtr, th dsigr must at last spcify four paramtrs to complt th spcificatio Pass bad cut off frqucy c Stop bad frqucy h Pass bad rippl δ Stop bad attuatio δ A similar st of paramtrs ar usd to spcify th othr typs of filtrs Th Phas Rspos is typically ot spcifid, xcpt, prhaps, wh a liar phas rspos is rquird. ELEC97

13 Spcificatio of a Digital filtr Th dsird magitud spcificatio for a low pass filtr ca th b writt as δ δ δ Pass bad Stop bad Th frqucy rag from c to h is ow as th trasitio bad, which is ffctivly a do t car rgio, i trms of th dsird magitud rspos A st of mappig ruls xists to trasform a digital low pass filtr ito a quivalt vrsio of ay of th othr thr typs of digital filtrs c, typically, a quivalt digital low pass filtr is first dsigd ad th trasformd ito a digital high pass filtr h c

14 Spcificatio of a Digital filtr δ -δ Pass Bad Trasitio Bad Stop Bad δ c h ELEC97

15 Filtr Dsig Filtr dsig pacags utiliss th frqucy magitud rspos providd PLUS a idicatio of whthr a IIR, or FIR, filtr is rquird Dsig tchiqus provid th lowst ordr complxity filtr such that its frqucy rspos is withi th providd magitud rspos spcificatio For a giv magitud rspos th ordr of a FIR filtr, which will mt this spcificatio, will b substatially highr tha th ordr of a IIR filtr, implmtig th sam magitud rspos But, FIR is rquird if liar phas is a cssity! ELEC97

16 FIR Filtrs Systm fuctio cotai oly ros o-rcursiv or rcursiv structurs ar both possibl; th bst ow is th o-rcursiv trasvrsal structur. FIR Filtrs ca hav a xactly liar phas rspos. Th implicatio of this is that o phas distortio is itroducd ito th sigal by th filtr. Th ffcts of usig a limitd umbr of bits to implmt filtrs such as roud off ois ad quatiatio rrors ar much lss svr i FIR tha i IIR. FIR rquirs mor cofficits for sharp cut-off filtrs tha IIR. Thus for a giv amplitud rspos spcificatio, mor procssig tim ad storag will b rquird for FIR implmtatio. Complxity is proportioal to th lgth of th impuls rspos. IIR Filtrs Cotai pols ad ros ormally Oly rcursiv structur is possibl; th most widly usd form is th cascad coctio of first-ordr ad scod ordr sctios. Th phas rsposs of IIR filtrs ar oliar, spcially at th bad dgs. Bcaus of quatiatio of th filtr cofficits, a pol ca i pricipl mov from a positio isid th uit circl to a positio outsid th uit circl ad hc caus istability. IIR rquirs fwr cofficits for sharp cut off filtrs tha FIR. o dirct rlatio btw th complxity ad th lgth of th impuls rspos which is ifiit by dfiitio. FIR filtrs hav o aalogu coutrpart. FIR Aalogu filtrs ca b radily trasformd ito dsig procdurs ar ormally itrativ quivalt IIR digital filtrs mtig similar procdurs. Dsig quatios do ot xist. spcificatios. IIR filtrs ca b dsigd usig 6 dsig formula.

17 Exrcis: Th followig trasfr fuctios rprst two diffrt filtrs mtig idtical amplitud frqucy rspos spcificatios. Dtrmi ad commt o th computatioal ad storag rquirmts. a a a Filtr : b b whr a.49889; a ; a.49889; b ; b ; h h h Filtr : h [ ] [ ] h[ ] [ ] h [ ] [ ] h[ 9] h h h [ 3] h[ 8] [ 4] h[ 7] [ 5] h[ 6] 7

18 As: FIR IIR Filtr Filtr umbr of Multiplicatios 5 umbr of additios 4 Storag locatios cofficits ad data ELEC97

19 Filtr Implmtatio Oc a suitabl filtr has b dsigd i.. th diffrc quatio dtrmid, it is cssary to cosidr issus cocrig its implmtatio Softwar or ardwar Fixd or Floatig Poit umbr of bits pr sampl, cofficit, multiplir ad accumulator Paralll or Cascad 9 ELEC97

20 Filtr Dsig Tchiqus IIR filtrs hav a quivalt aalogu filtr. c, th most commo mthod for dsigig IIR filtrs is, To fid a suitabl aalogu filtr ad th to trasform it ito a quivalt digital filtr FIR filtrs do OT hav a quivalt i th aalogu domai. All FIR filtr dsig tchiqus ar itrativ procdurs. ELEC97

21 IIR Filtr Dsig Tchiqus O dirct mthod for dsigig a simpl IIR filtr or idd a simpl FIR is ow as Pol-Zro Placmt If a IIR filtr is to b dsigd to a rasoabl dgr of accuracy i trms of its dsird frqucy rspos, th pol-ro placmt is OT a accptably accurat tchiqu Two commoly usd idirct dsig tchiqus for IIR filtrs ar Impuls Ivariat Trasformatio Biliar Trasformatio ELEC97

22 Trasformatio Tchiqus Both tchiqus ca trasform a suitabl stabl aalogu trasfr fuctio s ito a stabl digital trasfr fuctio, Both tchiqus us a particular mappig for th LaPlac variabl s i s ito th trasform variabl i Th trasformatio usd must maitai th sstial proprtis i.. stability, filtr typ, cut off frqucy tc. of th aalogu filtr s frqucy magitud rspos ELEC97

23 Impuls Ivariat Trasformatio Th impuls ivariat trasformatio dsigs a digital filtr such that its impuls rspos, h, is a sampld vrsio of th cotiuous impuls rspos of th aalogu filtr, ht I othr words, hht whr T: Samplig Priod This, howvr, dos OT imply that th rsultig digital filtr has th sam frqucy rspos as th origial aalogu filtr ovr th frqucy rag of to f samplig /, i.. << ELEC97

24 Aliasig Effcts All aalogu filtrs hav a frqucy magitud rspos which must hav a o-ro valu at ALL frqucis Wh th cotiuous impuls rspos of th aalogu filtr is sampld, th rsultat digital filtr s frqucy rspos is that of th aalogu filtr REPEATED priodically at a itrval qual to th samplig frqucy c, if th valu of th aalogu filtr s magitud rspos is sigificatly larg at AY frqucy, gratr tha half th samplig frqucy, this WILL rsult i sigificat aliasig of th origial aalogu filtr s frqucy rspos ELEC97

25 Aliasig Effcts c, for xampl, this tchiqu CAOT b usd to trasform a high pass aalogu filtr ito a high pass digital filtr Th samplig priod must b carfully slctd to b high ough to sur that this aliasig ffct has a miimal distortio of th origial dsird aalogu filtr s frqucy rspos Th lsso: Similar bhaviour i th TIME domai for aalogu ad digital filtrs, DOES OT sur similar bhaviour i th FREQUECY domai. 5 ELEC97

26 ELEC97 6 Mappig Rul Cosidr a simpl aalogu low pass filtr with a cut off frqucy of ω c radias s - Th impuls rspos of this filtr is a > c a s s ω t t < t a c t h ω

27 Mappig Rul If w sampl this sigal, with a samplig priod T, to produc th impuls rspos of th digital filtr, w gt ω c T h < This squc has th trasform T 7 ω c ELEC97

28 Mappig Rul Thus th mappig rul to gt from aalogu filtr, s, to quivalt digital filtr,, is giv by ω s ω ct c ot that thr is O rul for mappig ros of th aalogu filtr Oly possibl to dsig crtai typ of low pass filtrs usig this trasformatio 8 ELEC97

29 Trasformatio Clarly, from th mappig rul, a pol of th aalogu filtr at s-ω c is mappd ito a pol of th digital filtr at - ω ct Th trasformatio that has b carrid out is st ; This trasformatio maps th lft half sid of th s pla, i strips /T wid, isid th uit circl o th pla Stabl aalogu filtr bcoms a stabl digital filtr Aliasig occurs bcaus sparat pols i th s pla ca map ito th sam pol i th pla 9 ELEC97

30 Trasformatio Mappig ω 3/T s-pla -pla /T /T σ - -3/T Each strip maps oto th itrior of th uit circl 3 ELEC97

31 Exampl Dtrmi th trasfr fuctio of a low pass filtr, usig th impuls ivariat trasformatio, of th followig aalogu filtr s s 4 s Suggst a suitabl valu for th samplig priod T which should b usd ad, hc, dduc th diffrc quatio for th filtr 3 ELEC97

32 ELEC97 3 Exampl 3T T T T T T T Us trasformatio o EAC trm s 4 s 4 s 4 B A -4 4 ad B A B A s s B Bs A As s B s A s s 4 s

33 Exampl For a trm /sa, a suitabl samplig frqucy would b at fiv to t tims th cut off frqucy of th aalogu filtr I this problm, th filtr is formd as two such trms, hc, w should choos th o with th highr cut off frqucy I this problm, ω c radias s - or / c, a suitabl samplig frqucy f s would b 5/, which is a samplig priod of /5 scods 33 ELEC97

34 Exampl Trasfr fuctio bcoms: 4 T T T 3T.9955 With T, Diffrc Equatio : T y[].9955x[ -].88y[ -]-.58y[ - ] 34 ELEC97

35 Exrcis Usig impuls ivariat mthod dsig a digital filtr to approximat th followig ormalid aalogu trasfr fuctio: s s s Assum that th 3dB cut-off frqucy of th digital filtr is 5 ad th samplig frqucy is.8 35 ELEC97

36 ot: Bfor applyig th impuls ivariat mthod, w d to d-ormali th trasfr fuctio. As: 36 s s c s s s s s s ω ω ω

37 ELEC97 37 Exrcis Th followig is a -trasfr fuctio of a filtr dsigd usig th Impuls Ivariat Trasform, fid s. 3 3 T T T T

38 Biliar Trasformatio Biliar trasformatio is a far mor usful tchiqu as it dos ot hav th aliasig problms associatd with th impuls ivariat trasformatio Ths aros du to th fact that th impuls ivariat trasformatio mappd th s pla, i strips rathr tha i its tirty, isid th uit circl Th biliar trasformatio maps th tir lft had sid of th s pla ito th isid of th uit circl, i o pic 38 ELEC97

39 Trasformatio Mappig ω s-pla -pla σ Imag of th lft had s-pla. 39 ELEC97

40 ELEC97 or...!...! T s T s st st st st st st trms ordr th highr out droppig By st st st st st st st st Trasformatio Equatio

41 Frqucy Warpig Whil th biliar trasformatio dos ot suffr ay aliasig problms, it dos this at th xps of warpig th frqucy axis Th tir aalogu frqucy axis from ω qual to ro to ω qual to ifiity is mappd ito th digital filtr s frqucy rag of qual to ro to qual to Thrfor: ω digital digital aalogu aalogu ω ELEC97

42 Warpig Fuctio owvr, this warpig fuctio will mov all othr importat frqucy poits durig th trasformatio.g. cut off frqucis Without taig this warpig ito accout, a aalogu filtr with a cut off frqucy of f c will OT rsult i a digital filtr with a cut-off frqucy at f c /f s Thrfor, it is cssary to pr-warp ay cut-off frqucis, which must b maitaid accuratly, bfor a suitabl aalogu filtr is slctd This is do usig th Warpig fuctio rlatig digital frqucy to aalogu frqucy ELEC97

43 ELEC97 ta ta cos si mappd oto th uit circl is s W d to cosidr how th s domai frqucy axis T T is that T T T T T s d a warpd pr ω ω ω ω ω Warpig Fuctio

44 Warpig Fuctio Thus a o-liar rlatioship xists btw ω ad as show ωt - ELEC97

45 Practical Dsig Biliar trasformatio is th most commoly usd mthod Dsird magitud spctrum is ffctivly prwarpd ito th aalogu domai Suitabl aalogu filtr typ is slctd from wll ow classical aalogu filtr typs, ithr through dsig quatios or, mor commoly, tabls or filtr dsig hadboos: Buttrworth o rippl i pass or stop bad maximally flat frqucy rspos Chbyshv Typ : Rippl i pass bad, Typ : Rippl i stop bad Elliptical Lowst complxity but rippl i both pass ad stop bad

46 Exampl A simpl aalogu low pass filtr is giv by th trasfr fuctio s ωc s ω c This filtr has a 3 db cut off frqucy at ω c Dtrmi, usig th biliar trasformatio, a quivalt digital low pass filtr Th dsird cut off frqucy is ad th samplig frqucy of th systm is 8 46 ELEC97

47 Exampl Firstly, w caot simply us th biliar trasformatio o s ωc s ω This would rsult i a low pass filtr BUT, th cut off frqucy of th filtr would b warpd d which is OT what w wat ωd ta T ω 5987 rad s c ωat - s 6 ta 6 or f 95 d

48 ω Thrfor, th cut-off frqucy must b pr-warpd f f s prwarpd 8 4 Exampl ta 6 ta rad s T 8 Istad of it is ow 55 Thus th dsird aalogu filtr trasfr fuctio will b ωc s s ω s c 55

49 Exampl Thus th digital filtr s trasfr fuctio will bcom: Diffrc Equatio : y[].99x[].99x[ -].44y[ -]

50 Exampl A ormalisd Low Pass Filtr trasfr fuctio is giv blow. Dtrmi th -trasfr fuctio ad diffrc quatio usig Biliar Trasformatio. Assum fc 3 ad Fs 5 s s 5 ELEC97

51 Exampl Stp: Firstly, dtrmi th rquird digital cut-off frqucy c Stp: Th pr-warp ω 3 c ta 7.95rads c T / s Stp3: Dormalis filtr trasfr fuctio s s s s ω s c ω c ω c 5 ELEC97

52 Exampl Cotiud Stp4: Apply Biliar Trasformatio s s T ELEC97

53 Exrcis It is rquird to dsig a digital filtr to approximat th followig aalogu trasfr fuctio s s s usig th Bi-liar trasformatio mthod obtai th trasfr fuctio, s of th digital filtr assumig a 3dB cut-off frqucy of 5 ad a samplig frqucy of.8. As: ELEC97

54 Filtr Dsig: Pol-Zro Placmt Tchiqu Th positio of pols ad ros i th -pla hav a particular ffct o a filtrs frqucy rspos summarisd as follows: Zros placd ar th uit circl rsult i a otch i th frqucy spctrum Zro placd o th uit circl producs complt rctio of a siusoid at that frqucy Pols placd clos to th uit circl giv ris to larg pas at th associat frqucy By stratgically placig pols ad ros w ca arriv at a particular rspos Tchiqu caot b usd to dsig vry accurat filtrs

55 Exampl Stch th Frqucy Rspos of th systm dscribd by th giv pol-ro plot uit circl -pla 55 ELEC97

56 Frqucy Rspos Amplitud fs/4 fs/ 3fs/4 fs / frqucy 56 ELEC97

57 Dsig Exampl Dsig a filtr usig th pol-ro placmt tchiqu which has th followig spcificatio Complt sigal rctio at DC arrow pass bad at 5 Pass badwidth Samplig frqucy of 5 57 ELEC97

58 Dsig Solutio Complt Sigal Rctio at DC implis d for a ro at > Rcall scod ordr rsoat systm This systm ca b usd to gt a basic bad pass filtr Badwidth is giv by f b b r r cos r f f samplig b b 58 ELEC97

59 ] [.8783 ] [.579 ] [ ] [ ] [ y y x x y Diffrc Equatio ro trm is Zro at rcos b f f r b f f r Pol Trm samplig samplig

60 6 Proy s mthod ] [ m m m M i i i h a b Apart from dsigig IIR filtr from aalogu prototyps, it could b dsigd by miimisig a rror critrio i tim domai. For a IIR filtr with a dsird ad actual impuls rsposs of h d [] ad h[] th last squar rror critrio is Furthr th h[] could b writt as ] [ ] [ d h h E

61 Proy s mthod Accordig to Proy s algorithm first th domiator cofficits, a m ar foud by miimisig th rror. If th iput to th systm is δ[] th a stimat of h[] could b foud for >M as [ ] hr a ar foud by miimisig th followig rror hˆ d a h d [ ] Error M h d [ ] hˆ d [ ] Th th umrator cofficits, b i ar foud by matchig th dsird impuls rspos for a lgth. Matlab commad : proy 6

62 Sha s mthod I this mthod th last squar approach is usd to fid both umrator ad domiator cofficits M sparatly. b If th iput to th systm is δ[] th a stimat of h[] could b foud for >M as hˆ d [ ] a h d First stp: ths domiator cofficits, a, ar foud by miimisig th followig last squar rror this stp is similar to Proy s mthod. Error M h d [ ] hˆ [ ] d 6 [ ] a

63 Scod stp: th cosidr a filtr structur as blow whr a ar th cofficits foud prviously. δ[] Sha s mthod aˆ r b will b foud by miimisig th followig last squar rror. Error h d M whr ad h d [ ] hˆ d υ[] [ ] M b ˆ [ ] [ ] bυ[ ] υ[ ] δ[ ] aˆ υ[ ] h d 63

64 Digital to Digital Trasformatios Low pass to low pass si[ p p ' / ] α α si[ p p ' / ] α whr p & p ' ar th origial & w cut-off frqucis Low pass to high pass α α α cos[ cos[ p p ' / ] p ' / ] p cos[ u L / ] α α cos[ u L / ] α cot u L / ta Low pass to bad pass p l whr & u ar th lowr & uppr cut-off frqucis 64

65 Digital to Digital Trasformatios Low pass to bad stop 65 α α

66 FIR filtr dsig mthods Widowig mthod Frqucy samplig mthod Pars McLlla mthod Last squar mthod 66 ELEC97

67 Widowig mthod Th asist way to obtai a FIR filtr is to simply trucat th impuls rspos of a IIR filtr. h [ ] hd [ ] whr h d [] is th impuls rspos of th dsird IIR filtr, ad h[] is th FIR filtr. I gral h[] ca b thought of as big formd by th product of h d [] ad a widow fuctio w[] as follows: h othrwis [ ] h [ ] w[ ] d W d 67 ELEC97

68 Effct of rctagular widow for a low pass filtr dsig / Th covolutio producs a smard vrsio of th idal low pass frqucy rspos d. Th widr th mai lob of W, th mor spradig, whr as th arrowr th mai lob largr th closr coms to d. Thr is a trad off of maig larg ough so that smarig is miimid, yt small ough to allow rasoabl implmtatio.

69 Som widow fuctios 69

70 Dsig Procdur First, th dsird impuls rspos d to b drivd DTFT of th rquird filtr Th th dsird impuls rspos d to b multiplid by a appropriat widow fuctio. 7 ELEC97

71 For a idal low pass filtr with liar phas of slop - β ad cutoff ω c ca b charactrid i th frqucy domai by d β w c w ω c < ω Thus th corrspodig impuls rspos obtaid by DTFT is, h d [ ] si [ ω ] c β β A causal FIR filtr with impuls rspos h[] ca b obtaid by multiplyig by a widow bgiig at th origi ad dig at - as follows si [ ] [ ωc β ] h ω[ ] β For h[] to b a liar phas, β must b slctd so that th rsultig h[] is symmtric 7 β

72 Exrcis: a Dtrmi th impuls rspos of th lowpass filtr whos magitud rspos is giv by 3 d b To obtai a fiit impuls rspos from a rctagular widow of lgth 9 is usd. Comput th cofficits of th FIR filtr with a liar phas charactristic ad with this fiit impuls rspos. As: a 3 h d [ ] si / 3 As: b <

73 73 [ ] si for d h d d < 3 3 [ ] / 3 si 3 h d To b a liar phas filtr it should b symmtric about 4 9

74 [ ] h [ ] [ ] h d ω h h h h h [ ] [ ] [ ] [ 3] [ 4] Basd o th symmtry th cofficits ar shiftd ad flippd:

75 75 [ ] β β β for d h d d si 3 < β 3 3 [ ] β β β β h d 3 / si β Altrativly you ca icorporat th phas wh calculatig th impuls rspos as blow,

76 [ ] h [ ] [ ] h d ω h h h h h [ ] h[ 8] [ ] h[ 7] 3 4 [ ] h[ 6] [ 3] h[ 5] [ 4]. 333 Basd o th symmtry th cofficits ar shiftig ad flippig:

77 Frqucy-Samplig Filtr Aothr typ of FIR filtr dsig which uss th frqucy sampld valus of th dsird filtr rspos. Dsird filtr Frqucy samplig th filtr Th filtr is implmtd usig th frqucy sampld valus. 77 ELEC97

78 ELEC97 78 Frqucy-Samplig Filtr Although it was implid bfor that all FIR filtrs ar o-rcursiv this is ot strictly tru a part of th FIR filtr ca hav rcursiv structur, providd that th ovr all filtr dos ot hav ay pols Cosidr th followig FIR filtr Th corrspodig filtr systm fuctio is othrwis g h [ ]... p p g g

79 g g Comb Filtr c Rsoator R This ca b implmtd i stags as follows: x / g y - - COMB FILTER -. RESOATOR

80 Exampl 8, g 8 8 x / x / /8 g y / Filtr implmtd usig Comb filtr rsoator structur y /

81 Aalysis of c i Frqucy Domai c ; c si c is show o xt slid for 8. c has ulls qually spacd i th rag corrspodig to ros Shap of magitud rspos givs this filtr its am comb filtr 8 ELEC97

82 /4 Comb Filtr Frqucy Rspos 8 ros o th uit circl. c.5..5 /8 thta :pi/64: *pi; 8; mag abs/*si*thta/; plotthta,mag, axis tight;titl'comb Filtr Frqucy Rspos';

83 Pol-Zro Pattr for Comb Filtr c c Im{} Pol-Zro Pattr for Comb Filtr 8 3/ / /4 R{} uit circl This quatio xplicitly shows qually spacd ros aroud uit circl First ro at -3/ -/ -/4 8 pols

84 Stability of g g Comb Filtr c Rsoator R R has a pol o th uit circl at frqucy i.. at Sic pol is ot isid th uit circl filtr is ot stabl Th ro at of comb filtr cacls th rsoator pol at Th combiatio is thus STABLE

85 Ovrall Frqucy Rspos of Comb Filtr plus Rsoator si si si si ; g g g g

86 Ovrall Systm Frqucy Rspos has a maximum valu g at at / / / thta -pi:pi/64: pi; 8; g; A g/*si*thta/; B sithta/; mag absa./b; plotthta,mag, axis tight; titl'ovrall Filtr Frqucy Rspos';

87 Lt us cosidr a scod-ordr rsoator, whos cofficits ar ral-valud ad th pols ar situatd at th followig ro locatios of th comb filtr. 8 pols 8 3 / Im{} / /4 R{} uit circl -3 / - / - /4

88 Th systm fuctio of th scod-ordr rsoator ca b writt i paralll form as g g R T g Th abov procdur ca b gralisd to iclud ach scod-ordr rsoator cacllig a pair of ros of th comb filtr. Ths rsoators ar all coctd i paralll ad this paralll combiatio is coctd i cascad with th comb filtr to produc th total filtr T giv by:

89 ELEC97 Frqucy Samplig Filtr Ralisatio X / - - Y g g 4 4 g g g g g g g ad W ca show that

90 Frqucy Samplig Filtr Ralisatio ] [ ] [ ] [ ] [ it, trasform of Taig th - ] [ ] [ h Lt th dsird filtr rspos as d []. If this rspos is frqucy sampld at poits th th frqucy sampld valus ar d [] for,, -. Th th impuls rspos of th filtr h[] ca b xprssd as,

91 ] [ ] [ ] [ usig gomtric sris, ] [ ] [

92 Exampl: Lt us cosidr implmtig th liar itrpolator with th comb ad rsoator structur. Th impuls rspos of th liar itrpolator is giv by h ½; h ; h ½ ad h othrwis Solutio: [ cos ] Sic th umbr of lmts i th uit sampl rspos is qual to 3, w choos 3. Th comb filtr systm fuctio is th c 3 3 ELEC97

93 Im c 3 3 R c has thr ros locatd at /3 for, ad. Th ro at will b caclld by a ral rsoator, ad ros at ±/3 will b caclld by a pair of complx rsoators.

94 ; ; * ; ; g g g g R R ] cos [ Th gais of th rsoators ar qual to:

95 Comb ad Rsoator Structur X -3 X /3 - / y / 3 y - Comb / -/ Rsoator ELEC97

96 Exrcis: A frqucy-samplig filtr is show blow ad 3 a - - X - - / Y. a - / - a / - a b c Dtrmi a, a ad a - such that this filtr has a ral impuls rspos h, whr 3 ad Draw th frqucy-samplig filtr structur usig dlay lmts, multiplirs ad addrs. Driv ad xprssio for. Giv a filtr that has th sam frqucy rspos, but ralisd as FIR filtr.

97 Solutio: R Im ; 3 3 a a a ; ; R R 3 Part a:

98 } { R R Part c: h 5; h -4 ad h Th structur is similar to th prvious xampl

99 Advatags Vry simpl: Just sampl th dsird rspos. Rcursiv implmtatio of FIR filtrs gratly rducs th umbr of arithmtic opratios spcially multiplicatios Particularly attractiv solutio if w wat to ma a arrow-bad filtr fw o ro valus Frqucy samplig filtrs ormally hav small itgr cofficits. 99 ELEC97

100 Practical Cosidratios I thory w assum that pols ad ros coicid xactly o th uit circl This rquirs th filtr cofficits to b vry accurat % Th bst w ca do is to gt th pols i th viciity of ros This ca lad to a vry rratic local variatio of th frqucy rspos ELEC97

101 Practical cosidratios I practic, both ros ad pols ar dlibratly locatd ust isid th uit circl Th comb filtr systm fuctio is chos as a c whr a is mad slightly lss tha ELEC97

102 Computr-Aidd dsig of quirippl liar phas FIR filtr dsig I this dsig th obctiv is to miimis th wightd rror btw th dsird ad th actual amplitud rsposs. ε. W [ D ] Wh th pa absolut valu of th rror is miimisd miimax or Chbyshv critrio th rsultig FIR filtr is usually calld quirippl FIR filtr. Pa absolut rror is, ε max max ε

103 Par McCllla Mthod Par McCllla solvd this optimiatio for liar phas FIR filtrs ad that mthod is calld Par McCllla mthod. Matlab commad is FIRPM or rm 3 ELEC97

104 ELEC97 Last Squar Tchiqu Filtrs ca b dsigd basd o miimisig th last wightd squard rror. Th squard rror wh P is th wightig fuctio is; 4 ε d P d

105 If th filtr cofficits, h[] is {b } th W watd to fid th cofficits which could miimi th abov rror. So by taig partial drivativs with rspct to b ad quatig thm to ro, 5 ε d b b P d U L d U L * ε d b b P d U L d U L *

106 Basd o th symmtry proprty for ral cofficits: * - 6 b ε ε d b b P b d U L d U L * [ ] [ ] ε d P d P b b d d U L * ] [ ] [ d r b b U L ε [ ] d P d d P r d ] [ ] [

107 This is th simultaous quatios ad i matrix form 7 ] [ ] [ d r b b U L ε ] [. ] [ ] [. [] ] [ ] [.... ] [... [] ] [ ] [... [] [] U d L d L d b b b r U L r U L r L U r r r L U r r r U L L br - d Thus th filtr cofficits, b ca b foud by solvig ths quatios.

108 Summary Spcificatios of a digital filtr IIR filtr dsig tchiqus Trasform mthods from aalogu prototyps Biliar trasform Impuls ivariat trasform Pol ro placmt Proy s mthod Sha s mthod FIR filtr dsig tchiqus Widowig mthod Frqucy samplig mthod Pars McLlla mthod Last squar mthod 8 ELEC97