Digital Signal Prossing, Fall 006 Ltur 7: Filtr Dsign Zhng-ua an Dpartmnt of Eltroni Systms Aalborg Univrsity, Dnmar t@om.aau. Cours at a glan MM Disrt-tim signals an systms Systm MM Fourir-omain rprsntation Sampling an ronstrution Systm analysis MM5 Systm strutur MM6 MM4 Filtr sign -transform DF/FF MM7, MM8 MM3 MM9, MM0
Part I: Filtr sign Filtr sign IIR filtr sign Analog filtr sign IIR filtr sign by impuls invarian IIR filtr sign by bilinar transformation 3 Filtr sign pross Filtr, in broar sns, ovrs any systm. hr sign stps Problm Spifiations Prforman onstraints Approximations Systm funtion raliation Solution Magnitu rspons Phas rspons (frquny omain Complxity IIR or FIR Subtyp Strutur 4
Spifiations an xampl Spifiations for a isrt-tim lowpass filtr 0.0 ( ( 0.00, + 0.0, ω ω s 0 ω ω p δ 0.0 δ 0.00 5 Spifiations of frquny rspons ypial lowpass filtr spifiations in trms of tolrabl Passban istortion, as smallst as possibl Stopban attnuation, as gratst as possibl With of transition ban: as narrowst as possibl Improving on oftn worsns othrs a traoff Inrasing filtr orr improvs all 6 3
D filtr for C signals Disrt-tim filtr for th prossing of ontinuoustim signals Banlimit input signal igh nough sampling frquny hn, spifiations onvrsion is straightforwar jω (, Ω < π / ff ( jω 0, Ω > π / ω ( ff ( j, ω < π ω Ω Fig. 7. 7 Spifiations an xampl Spifiations for a ontinuous-tim lowpass filtr 0.0 ff ff ( jω + 0.0, 0 Ω π (000 ( jω 0.00, Ω π (3000 Fig 7.(a(b δ 0.0 δ 0.00 Ω Ω p s π (000 π (3000 8 4
Dsign a filtr Dsign goal: fin systm funtion to ma frquny rspons mt th spifiations (tolrans Infinit impuls rspons filtr Pols insir unit irl u to ausality an stability Rational funtion approximation Finit impuls rspons filtr Linar phas is oftn rquir Polynomial approximation 9 E.g. IIR filtr sign For rational systm funtion ( fin th systm offiints suh that th orrsponing frquny rspons ( ( provis a goo approximation to a sir rspons ( ( sir M 0 N b a ( Rational systm funtion Stabl ausal 0 5
FIR or IIR Eithr FIR or IIR is oftn pnnt on th phas rquirmnts Only FIR filtr an b at th sam tim stabl, ausal an GLP If ( is stabl an GLP, any non-trivial pol p insi th unit irl orrspons a pol /p outsi th unit irl, so that ( annot hav a ausal impuls rspons (as ROC is a ring inluing unit irl. Dsign prinipl If GLP is ssntial FIR If not IIR prfrabl (an mt spifiations with lowr omplxity FIR an IIR IIR Rational systm funtion Pols + ros Stabl/unstabl ar to ontrol phas Low orr (4-0 Dsign on th basis of analog filtr FIR Polynomial systm funtion Zros Stabl Easy to gt linar phas igh orr (0-000 Unrlat to analog filtr 6
Part II: IIR filtr sign Filtr sign IIR filtr sign Analog filtr sign IIR filtr sign by impuls invarian IIR filtr sign by bilinar transformation 3 Dsign IIR filtr bas on analog filtr h mapping is irt jω (, ff ( jω 0, ω ( ff ( j, Ω < π / Ω > π / ω < π Avan analog filtr sign thniqus Dsigning D filtr by transforming prototyp C filtr: ransform (map D spifiations to analog Dsign analog filtr Invrs-transform analog filtr to D 4 7
ransformation mtho ransform (map D spifiations to analog ω Ω Dsign analog filtr ( s or h ( t Invrs-transform to D ( or h[ n] h imaginary axis of th s-plan th unit irl of th -plan Pols in th lft half of th s-plan pols insi th unit irl in th - plan (stabl s σ + jω ( s r X ( ( jω X ( n n h( t st h( t x[ n] x[ n] jωt n t t n 5 Part III: Analog filtr sign Filtr sign IIR filtr sign Analog filtr sign IIR filtr sign by impuls invarian IIR filtr sign by bilinar transformation 6 8
Analog filtr sign Buttrworth Chbyshv I Chbyshv II Ellipial 7 Buttrworth lowpass filtrs h magnitu rspons Maximally flat in th passban Monotoni in both passban an stopban h squar magnitu rspons ( jω N + ( Ω / Ω FigB.. 8 9
Pols in s-plan 9 Chbyshv filtrs Chbyshv II filtrs: quirippl in passban, flat in th stopban Chbyshv II filtrs: quirippl in stopban, flat in th passban 0 0
Ellipti filtrs Equirippl both in stopban an in th passban Part IV: Dsign by impuls invarian Filtr sign IIR filtr sign Analog filtr sign IIR filtr sign by impuls invarian IIR filtr sign by bilinar transformation
Filtr sign by impuls invarian Impuls invarian: a mtho for obtaining a D systm whos ( is trmin by th ( jω of a C systm. h [ n] h ( n - 'sign' sampling intrval In D filtr sign, th spifiations ar provi in th isrt-tim, so has no rol. is inlu for isussion though. also has nothing to o with C/D an D/C onvrsion in Fig. 7. 3 Rlationship btw frquny rsponss Impuls rspons sampling: h [ n] h ( n Frquny rspons ω π ( ( j + j if th C filtr is banlimit ( jω 0, Ω π / thn ( ω ( j, ω π his is also th way to gt C filtr spifiations from ( by applying th rlation Ω ω / 4 x[ n] x ( n ω π X ( X ( j j
3 5 Aliasing in th impuls invarian sign ( ( j j j π ω ω + ( ] [ n h n h 6 Rlationship btw systm funtions h transform from C to D is asy to arry out as a transformation on th systm funtion Rational systm funtion, aftr partial fration xpansion N n s N n s n u A n u A n h n h ] [ ( ] [ ( ] [ < 0 0, 0, ( ( t t A t h s s A s N t s N N s A (
Impuls invarian with a Buttrworth filtr Spifiations 0.895 ( (, 0 ω 0.π 0.7783, 0.3π ω π Sin th sampling intrval anls in th impuls invarian prour, w hoos, so ω Ω Magnitu funtion for a C Buttrworth filtr 0.895 ( jω, 0 Ω 0.π ( jω 0.7783, 0.3π Ω π Du to th monotoni funtion of Buttrworth filtr ( j0.π 0.895 ( j0.3π 0.7783 7 Impuls invarian with a Buttrworth filtr ( (3 Squar magnitu funtion of a Buttrworth filtr ( Ω j N + ( Ω / Ω 0.π + ( Ω 0.3π + ( Ω N N N 5.8858 Ω ( ( 0.70474 0.895 0.7783 (4 (5 ( ( s N 6 Ω ( j0.π 0.895 ( j0.3π 0.7783 0.703 ( s + ( s / jω + ( s / j0.703 N 8 4
Impuls invarian with a Buttrworth filtr pols for th squar magnitu funtion h systm funtion has th thr pol pairs in th lft half of th s-plan 9 Impuls invarian with a Buttrworth filtr s ( s + 0.3640s + 0.4945( s 0.093 + 0.9945s + 0.4945( s ( 0.87 0.4466 ( (.97 + 0.6949.48 +.455 + (.069 + 0.3699.8557 0.6303 + ( 0.997 + 0.570 +.3585s + 0.4945 30 5
Impuls invarian with a Buttrworth filtr 3 Part V: Dsign by bilinar transformation Filtr sign IIR filtr sign Analog filtr sign IIR filtr sign by impuls invarian IIR filtr sign by bilinar transformation 3 6
Bilinar transformation By using impuls invarian, th rlation btwn C an D frquny is linar (xpt for aliasing, thus th shap of th frquny rspons is prsrv. But only propr for banlimit filtrs, problm for.g. highpass Bilinar transformation btwn s an s ( ( [ ( ] + + Invrs + ( ( / s / s 33 Bilinar transformation Givn s σ + jω if s jω + ( ( if σ < 0, if σ > 0, / s + σ / s σ < for any > for any / / Ω Ω + jω jω / / + jω jω so,, on th jω - axis i.. th / / for any s jω - axis maps onto th unit irl 34 7
Bilinar transformation frquny rlationship Consir frquny s ( + + s s jω jω jω jω ( + Ω tan( ω / ω artan( Ω [ / / / ( j sinω / j ] tan( ω / (osω / 35 Bilinar transformation h bilinar transformation maps th ntir jω -axis in th s-plan to on rvolution of th unit irl in th -plan. Compar with Ω ω / 36 8
Bilinar transformation of a Buttrworth filtr Spifiations 0.895 ( (, 0 ω 0.π 0.7783, 0.3π ω π Magnitu funtion for a C Buttrworth filtr 0.π 0.895 ( jω, 0 Ω tan( 0.3π ( jω 0.7783, tan( Ω Du to th monotoni funtion of Buttrworth filtr Choos ( j tan(0.π 0.895 ( j tan(0.5π 0.7783 37 Bilinar transformation of a Buttrworth filtr Squar magnitu funtion of a Buttrworth filtr ( Ω j N + ( Ω / Ω N 5.305 ( j tan(0.π 0.895 ( j tan(0.5π 0.7783 N 6 Ω 0.766 38 9
Bilinar transformation of a Buttrworth filtr 39 Summary Filtr sign IIR filtr sign Analog filtr sign IIR filtr sign by impuls invarian IIR filtr sign by bilinar transformation 40 0
Cours at a glan MM Disrt-tim signals an systms Systm MM Fourir-omain rprsntation Sampling an ronstrution Systm analysis MM5 Systm strutur MM6 MM4 Filtr sign -transform DF/FF MM7, MM8 MM3 MM9, MM0 4