Design of Infinite Impulse Response (IIR) digital filters
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1 Dign of Infinit Imul Ron (IIR) digital filtr Outut from a digital filtr i mad u from rviou inut and rviou outut, uing th oration of convolution: wo convolutionar involvd: on with th rviou inut, and on with th rviou outut. In ach ca th convolving function i calld th filtr cofficint. If uch a filtr i ubjctd to an imul (a ignal coniting of on valu followd by ro) thn it outut nd not ncarily bcom ro aftr th imul ha run through th ummation. So th imul ron of uch a filtr can b infinit in duration. Such a filtr i calld an Infinit Imul Ron filtr or IIR filtr. Not that th imul ron nd not ncarily b infinit: if it wr, th filtr would b untabl. In fact for mot ractical filtr, th imul ron will di away to a ngligibly mall lvl. On might argu that mathmatically th ron can go on for vr, gtting mallr and mallr: but in a digital world onc a lvl gt blow on bit it might a wll b ro. h Infinit Imul Ron rfr to th ability of th filtr to hav an infinit imul ron and do not imly that it ncarily will hav on: it rv a a warning that thi ty of filtr i ron to fdbac and intability.
2 h filtr can b drawn a a bloc diagram: officint calculation mthod for IIR filtr Whil trying to comut th filtr cofficint on ha to lct a mthod from a numbr of aroximation mthod in ordr to calculat th valu of d and c. A iml way to obtain th IIR filtr cofficint i to lac ol and ro judiciouly in th -lan uch that th rulting filtr ha th dird frquncy ron. hi aroach i nown a th ol-ro-lacmnt mthod. I only uful for vry iml filtr, i.. notch filtring, whr th filtr aramtr (aband ril) nd not b cifid rcily. A mor fficint aroach i firt to dign an analog filtr atifying th dird cification and thn to convrt it into an quivalnt digital filtr. Mot IIR digital filtr ar dignd thi way. hr ar thr main mthod for convrting an analog filtr into a digital filtr. ) imul invarianc ) th matchd -tranform 3) bi-linar -tranform Baic onct and illutrativ dign xaml: Whn a ro i lacd at a oint on th -lan, th frquncy ron will b ro at th corronding oint. A ol on th othr hand roduc a a at th corronding frquncy oint. Pol that ar clo to th unit circl giv ri to larg a whra ro clo to or on th circl through or minima. nc by tratgically lacing ol and ro on th -lan w can obtain iml lowa or othr frquncy lctiv filtr.
3 An imortant oint to rmmbr i that : In ordr for th filtr cofficint to b ral, th ol and ro mut b ithr ral (i.. li on th oitiv or ngativ ral axi) or occur in comlx conjugat air. F /4 F / 0 3F /4 0 F /4 F / 3F /4 Frquncy (a) (b) (a) Pol ro diagram of a iml filtr (b) A tch of it frquncy ron Examl: Illutrating th iml ol-ro mthod of calculating th filtr cofficint a banda digital filtr i rquird to mt th following cification: (i) (ii) (iii) comlt ignal rjction at D and 50. A narrow aband cntrd at 5 A 3dB bandwidth of 0. Auming a amling frquncy of 500, obtain th tranfr function of th filtr by uitably lacing -lan ol and ro, and it diffrnc quation. Solution: Sinc a comlt rjction i rquird at ro and 50 w nd to lac ro at th corronding oint on th -lan. h ar at angl of 0 and /50080 on th unit circl. o hava th aband cntrd at 5 rquir u to lac ol at ±360 5/500±90. o nur that th cofficint ar ral, it i ncary to hav comlx conjugat ol air.
4 Im R (a) (b) (a) Pol-ro dıagram (b) Bloc diagram rrntation of filtr h radiu of th ol i dtrmind by th dird bandwidth. An aroximat rlationhi btwn r, for r> 0.9, and bandwidth bw i: r ( bw / )π for our cification bw 0 and F500. hi lad to an r valu of r F [] ( )( ) jπ / jπ / ( r )( r ) h diffrnc quation i y [] n y[ n ] x[] n x[ n ] b 0, b 0, b - and a 0, a
5 Examl : (Uing th ol-ro lacmnt mthod to calculat cofficint of a notch filtr) Obtain by th ol-ro lacmnt mthod th tranfr function and th diffrnc quation of a iml digital notch filtr that mt th following cification. Solution: Notch frquncy 50 3dB width of notch ±5 amling frquncy 500 o rjct th comonnt at 50 w lac a air of comlx ro at oint on th unit circl corronding to 50 that i at angl of /500±36. o achiv a har notch filtr and imrovd amlitud ron on ithr id of th notch frquncy, a air of comlx conjugat ol ar lacd at a radiu r <. h width of th notch i dtrmind by th location of th ol. h rlationhi btwn th bandwidth and radiu i am a in th rviou xaml. Im [ f ] R Frquncy h rlationhi btwn th bandwidth and th rdiu i alicabl. hu th radiu of th ol i From th tranfr function of th filtr w hav: j36 j36 [] [ ][ ] [ ][ ]
6 h diffrnc quation will b: y [] n x[] n.680x[ n ] x[ n ].564y[ n ] y[ n ] b 0, b -.680, b and a -.564, a Imul Invariant Mthod of officint alculation In thi mthod, tarting with a uitabl analog tranfr function () th imul ron h(t) i obtaind uing th Lalac tranform. h h(t) o obtaind i uitably amld to roduc h(n), and th dird tranfr function () i thn obtaind by -tranforming h(n) whr i th amling intrval. Examl: (thortic) Digiti an analog filtr uing imul invarianc mthod. h analog filtr ha a tranfr function givn by: () Solution: If w aly th invr Lalac tranform w can gt h(t) a blow: h ( t) L t { ( ) } L According to th imul invariant mthod, th imul ron of th quivalnt digital filtr, h(n), i qual to h(t) at th dicrt tim tn, n 0,,,... n ( n ) h( t) h t n h tranfr function of [] i obtaind by -tranforming h(n) : n n 0 n 0 [] h( n ) n n hu from th rult o far w can writ : o aly th imul invariant mthod to a high ordr IIR filtr with iml ol, th tranfr function () i firt xandd uing artial fraction a th um of ingl-ol filtr:
7 M M M... ) ( whr ar th ol of ().Sinc ach trm ha th am form thn w can ay that: M M igh ordr IIR filtr ar normally ralid a cacad or aralll combination of tandard cond ordr filtr ction. For ca M ( ) ( ) ( ) If th ol and ar comlx conjugat th and will alo b comlx conjugat ( ) ( ) [ ] ( ) * co in co * i i i i r r r r r r r and i ar th ral and imaginary art of and r and i ar th ral and imaginary art of. Examl: Dign a digital filtr to aroximat th following normalid analog tranfr function: ) ( Uing th imul invariant mthod obtain th tranfr function, (), of th digital filtr, auming a 3dB cutoff frquncy of 50 and a amling frquncy of.8. Solution: Firt w nd to frquncy cal th normalid tranfr function. hi i achivd by rlacing by /α whr α π
8 ' () () α / α α α whr; ( j) * α ( j), α j j, Sinc th ol ar comlx conjugat, th tranformation i ud to obtain th dicrt-tim tranfr function, (). * r 0, i , i 0.507, r 0.507, r 0.594, in( ) , i co( ) i and r Uing th valu in th rviouly obtaind xrion w gt () a: [] If w ubtitut jw in th abov quation th valu of [] at w0 i 3, aroximatly qual to th amling frquncy. Such larg gain i charactritic of imul invariant filtr. o th gain down and to avoid ovrflow whn th filtr i imlmntd, it i common ractic to multily [] by (or quivalntly divid it by th amling frquncy) [] hrfor, w hav b 0 0 b a.0308 a
9 h bilinar tranformation Lt u conidr th idntity ( ) whr, and ar comlx variabl. hi i what i nown a th bilinar tranformation or utin tranformation. h invr of thi tranformation i aily dtrmind a jw In th dicrt domain w ar intrtd in valu on th unit circl whra in th Lalac domain valu on th imaginary axi (jω) i of imortanc. h tranformation and it invr ar wll dfind for all on th imaginary axi whr jω. r Ω dnot th analog frqunci in rad/c. h bilinar tranformation ti th two frquncy cal togthr a follow. Starting from ( ) jw and ltting w obtain: jω jw jw / jw / ( ) ( ) in( w/ ) jw jw / jw / i co ( w/ ) i tan( w/ ) and conquntly th rlation btwn th two frquncy cal i givn by Ω tan w ( / ) W not that w0 imli Ω0 and for incraing w th continuou frqunci will alo incra and a w π th Ω. nc th ntir continuou tim frquncy rgion Ω 0, i mad to th ur half of th unit circl. [ )
10 ranformd ranfr Function On imortant alication of th bilinar tranformation i for th convrion of an analog filtr to a digital on. h tranformation i alid a follow. Start from a Lalac domain filtr a () which i a rational function in th fr variabl. A rational tranfr function i th quotint of two olynomial. h bilinarly tranformd dicrt tim (Z-domain) tranfr function d [] i dfind a: d [] () ( ) [] a Not that d [] i alo a rational tranfr function. h MALAB command bilinar rform th bilinar tranformation. h following good rorti can b idntifid: It i ay to vrify that th numbr of ol of d [] and a () coincid. hi man that th ordr of th filtr i unchangd. If a () i a tabl tranfr function o i alo d [] h comlx filtr ron, i.. both magnitud and ha, coincid for frqunci rlatd by: a jw ( jω ) [ ] h lat fatur imli that th bilinar-tranformation rrv th ty of filtr. If a () i a low-a filtr thn d [] will alo b a low-a filtr. d d Analog IIR filtr dign o dign any of th four ty analog IIR filtr it i ufficint to ba th dign on an analog IIR lowa filtr. hi ction brifly ummari th dign of analog LP IIR filtr bad on filtr rototy. An analog rototy LP-filtr i givn by : LP ( ) n n ( )... n whr () i a rototy olynomial. h ty of LP filtr dfin th olynomial cofficint and th variou rototy olynomial can b found in tandard txtboo. h MALAB command butta and chba giv th ro and ol of th rototy filtr. Examl ar givn in th tabl blow:
11 Ordr Prototy olynomial () Gain abl : Buttrworth rototy olynomial Ordr Prototy olynomial () Gain abl : hbyhv ty rototy olynomial wit a-band ril 0.5dB. All rototy olynomial hav th 3dB cutoff frquncy normalid to Ωrad/. For a articular dign with a dird cut-off frquncy of Ω LPc th filtr i modifid by caling th variabl with th invr of th dird cut-off frquncy. h final analog dign i thn; LP ( ) LP ( ) n n Ω LPc... Ω LPc Ω LPc In MALAB ll rform thi dign from a givn rototy. n Ω LPc n P,BP and BS filtr dign hr othr ty of dign, high-a, band-a, and band-to can all b gnratd from a low-a filtr uing cial frquncy tranformation. W aum that LP () i a low-a filtr with -3dB cutoff frquncy Ω LPc and to band frquncy at Ω LP a illutratd in figur. Figur : h four tandard filtr : LP low-a, Phigh-a, BPband-a, BSband-to
12 igh-a dign A high-a dign i achivd from a low-a filtr by th tranformation P ( ) ( ) LP Ω I LP Ω I whr Ω Ω Pc P Ω Ω Ω Ω I LPc I LP r Ω I i a valid caling. In MALAB th function lh rform thi dign from a givn rototy. Analog IIR dign ummary Aum th rulting filtr ordr i givn. hi immdiatly contrain th width of th tranition on and w only hav to dign th oition of th cut-off frqunci. ) Idntify th cut-off frqunci (Ω XXlc and Ω XXhc ) of th dign cification. From thm dtrmin Ω and th cut-off frquncy Ω of th low-a filtr. I ) Dign a low-a filtr with th dird ΩLPc uing a low-a rototy filtr (.g. Buttrworth or hbyhv). If th final dign i a BP or BS filtr, thn thi low-a filtr hould hav half th ordr of th final dird dign. 3) U th corronding frquncy tranformation to finally obtain th dird filtr. LPc
13 Digital IIR filtr dign h dign of an IIR filtr i rathr traightforward whn uing th bilinar tranformation tchniqu. Mot of th wor i don whn digning th analog filtr. Rcall that if an analog filtr i convrtd to a digital on via th bilinar tranformation w obtain th rlation: A th bilinar tranformation rrv th ty of filtr (LP, BP tc.) w only nd to ma ur that th cut-off frqunci gt mad to th corrct lac. h dign tchniqu i ummarid hr:. Start by idntifying th dicrt tim frqunci dcribing th cutoff frqunci. U normalid frqunci in radian whr w π rrnt half th amlingfrquncy ( w πf/f ).. alculat th corronding analog frqunci uing Ω tan( w/ ) 3. Dign an analog filtr a () bad on th cification a dcribd in ction bfor. 4. U th bilinar tranformation to obtain th final digital (Z-domain) filtr.
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