Digital Signal Processing, Fall 2009

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1 Digital Signal Prossing, Fall 009 Ltur 4: Filtr Dsign Zhng-ua Tan Dpartmnt tof Eltroni Systms Aalborg Univrsity, Dnmark 1 Cours at a glan MM1 Disrt-tim signals an systms Systm MM Fourir transform an Z-transform Sampling an ronstrution Systm analysis DFT/FFT MM5 MM3 Filtr sign MM4 1

2 Part I: Filtr sign Filtr sign IIR filtr sign FIR filtr sign 3 Filtr sign pross Filtr, in broar sns, ovrs any systm. Thr sign stps Problm Spifiations Prforman onstraints Approximations Systm funtion Ralization Solution Magnitu rspons Phas rspons frquny omain Complxity IIR or FIR Subtyp Strutur 4

3 Spifiations an xampl Spifiations for a isrt-tim lowpass filtr , , s p Spifiations of frquny rspons Typial 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

4 DT filtr for CT signals DT filtr for th prossing of CT signals Banlimit input signal igh nough sampling frquny Thn, spifiations oftn givn in frquny omain onvrsion is straightforwar ff T, / T 0, / T T is small nough. ff, T T Fig. 7.1 Signal is ban-limit; 7 Spifiations an xampl Spifiations for a CT lowpass filtr , ff 0.001, 3000 ff T Fig 7.ab p s

5 Dsign a filtr Dsign goal: fin systm funtion to mak 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 M 1 k 0 z N b k z fin th systm offiints suh that th orrsponing frquny rspons z z provis a goo approximation to a sir rspons sir k 1 k a k z k z Rational systm funtion Stabl ausal 10 5

6 FIR an IIR IIR Rational systm funtion Pols + zros 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 Unrlat to analog filtr 11 FIR or IIR Whthr FIR or IIR oftn pns on th phas rquirmnts Dsign prinipl If GLP is ssntial FIR If not IIR prfrabl an mt spifiations with lowr omplxity 1 6

7 Part II: IIR Filtr sign Filtr sign IIR filtr sign Analog filtr sign IIR filtr sign by impuls invarian FIR filtr sign 13 Dsign IIR filtr bas on analog filtr Th mapping is irt T, / T ff 0, / T ff, T Avan analog filtr sign thniqus Dsigning DT filtr by transforming prototyp CT filtr: Transform map DT spifiations to analog Dsign analog filtr Invrs-transform analog filtr to DT Signal is ban-limit; T is small nough. 14 7

8 Transformation mtho Transform map DT spifiations to analog T Dsign analog filtr s or h t Invrs-transform to DT z or h[ Th imaginary axis of th s-plan th unit irl of th z-plan Pols in th lft half of th s-plan pols insi th unit irl in th z-plan stabl s s z r X X z h t n n st h t x[ x[ z t t n t n 15 Part II-A: Analog Filtr sign Filtr sign IIR filtr sign Analog filtr sign IIR filtr sign by impuls invarian FIR filtr sign 16 8

9 Analog filtr sign Buttrworth Chbyshv I Chbyshv II Ellipial 17 Buttrworth lowpass filtrs Th magnitu rspons Maximally flat in th passban Monotoni in both passban an stopban Th squar magnitu rspons N 1 /

10 Pols in s-plan 19 Part II-B: Dsign by impuls invarian Filtr sign IIR filtr sign Analog filtr sign IIR filtr sign by impuls invarian FIR filtr sign 0 10

11 Filtr sign by impuls invarian Impuls invarian: a mtho for obtaining a DT systm whos is trmin by th of a CT systm. h[ T T h nt - 'sign' sampling intrval In DT filtr sign, th spifiations ar provi in th DT, so T has no rol. T is inlu for isussion though. T also has nothing to o with C/D an D/C onvrsion in Fig. 7.1, i.. T n not b th sam as th sampling prio T of th C/D an D/C onvrsion. 1 Rlationship btw frquny rsponss Impuls rspons sampling: h[ T h nt Frquny rspons k T T if th CT filtr is banlimit 0, / T thn k, T x[ x nt 1 k X X T T T k ff, T This is also th way to gt CT filtr spifiations from by applying th rlation / T 11

12 Aliasing in th impuls invarian sign h[ T h nt k T T k Th ontinuous-tim filtr may b sign to x th spifiations, partiularly in th stopban. 3 Impuls invarian with a Buttrworth filtr Spifiations , , 0.3 Sin th sampling intrval T anls in th impuls invarian prour, w hoos T =1, so Magnitu funtion for a CT Buttrworth filtr , , 0.3 T, / T ff 0, / T ff, T Du to th monotoni funtion of Buttrworth filtr, w hav

13 Impuls invarian with a Buttrworth filtr Squar magnitu funtion of a Buttrworth filtr N 1 / N N N N s 1 s 1 s / 1 1 s / N 1 5 Impuls invarian with a Buttrworth filtr 1 pols for th squar magnitu funtion Th systm funtion has th thr pol pairs in th lft half of th s-plan 6 13

14 Impuls invarian with a Buttrworth filtr To onstrut s from th magnitu-squar funtion s-s, w hoos th pols on th lft-half-plan part of th s-plan to obtain a stabl an ausal filtr. s s s s s s z z z z s Exprss s as a partial fration xpansion, prform th transformation of Eq. 7.1: z z z z z 0.570z 7 Impuls invarian with a Buttrworth filtr 8 14

15 Part III: FIR filtr sign Filtr sign IIR filtr sign FIR filtr sign Commonly us winows Gnraliz linar-phas FIR filtr Th Kaisr winow filtr sign mtho 9 FIR filtr sign Dsign problm: th FIR systm funtion M z bk z k 0 k bn, 0 n M h[ 0, othrwis Start from impuls rspons irtly 1 z h[0] h[1] z... h[ M ] z M Fin th gr M an th filtr offiints h[k] to approximat a sir frquny rspons 30 15

16 Lowpass filtr as an xampl Ial lowpass filtr lp 1, 0, h [ n ] sinn, n n IIR filtr: bas on transformations of CT IIR systm into DT ons. 1 N 1 / pols FIR filtr: how? h[ is non-ausal, infinit! 31 Dsign by winowing Dsir frquny rsponss ar oftn piwisonstant with isontinuitis at th bounaris btwn bans, rsulting in non-ausal an infint impuls rspons xtning from to, but n, h [ 0 So, th most straightforwar mtho is to trunat th ial rspons by winowing an o timshifting: h [, n M g[ 0, othrwis h[ g[ n M ] 3 16

17 Dsign by winowing Aftr Champagn & Labau, DSP Class Nots Dsign by rtangular winow In gnral, h [ n ] h [ n ] w [ n ] For simpl trunation, th winow is th rtangular winow 1, 0 n M w[ 0, othrwis

18 Convolution pross by trunation Fig w[ 1, n, W 35 thn what? shoul b narrow ban r W r Rquirmnts on th winow r w [ 1, n, W r W shoul b narrow ban Rquirmnts W approximats an impuls to faithfully rprou th sir frquny rspons w[ as short as possibl in uration th orr of th filtr to minimiz omputation in th implmntation of th filtr Confliting Tak th rtangular winow as an xampl

19 Rtangular winow M+1 M=7 onstant 4 37 Part III-A: Commonly us winows Filtr sign IIR filtr sign FIR filtr sign Commonly us winows Gnraliz linar-phas FIR filtr Th Kaisr winow filtr sign mtho 38 19

20 Stanar winows tim omain Rtangular 1, 0 n M w[ 0, othrwis Triangular n / M, 0 n M / w[ n / M, M / n M 0, othrwis anning osn / M, 0 n M w[ 0, othrwis amming osn / M, 0 n M w[ Blakman 0, othrwis osn / M 0.08os4n / M, 0 n M w[ 39 0, othrwis Stanar winows figur Fig. 7.1 Plott for onvnin. In fat, th winow is fin only at intgr valus of n. 40 0

21 Stanar winows magnitu 41 Stanar winows omparison Magnitu of si lobs vs with of main lob Inpnnt of M! 4 1

22 Part III-B: Linar-phas FIR filtr Filtr sign IIR filtr sign Analog filtr sign IIR filtr sign by impuls invarian FIR filtr sign Commonly us winows Gnraliz linar-phas FIR filtr Th Kaisr winow filtr sign mtho 43 Linar-phas FIR systms Gnraliz linar-phas systm A A is a ral funtion of, an ar ral onstants Causal FIR systms hav gnraliz linar-phas if h[ satisfis th symmtry onition or h[ M h[, h[ M h[, n 0,1,..., M n 0,1,..., M 44

23 Gnraliz linar phas FIR filtr Oftn aim at signing ausal systms with a gnraliz linar phas stability is not a problm If th impuls rspons of th sir filtr is symmtri about M/, h [ M h [ Choos winows bing symmtri about th point M/ w[ M, 0 n M w[ 0, othrwis M / W W W is a ral, vn funtion of w th rsulting frquny rspons will hav a gnraliz linar phas A M / A is ral an vn 45 Linar-phas lowpass filtr an xampl Dsir frquny rspons so M /, lp 0, sin[ n M / ] hlp[, n n M / h [ M h [ lp lp apply a symmtri winow a linar-phas systm sin[ n M / ] h[ hlp[ w[ w[ n M / 46 3

24 4 Winow mtho approximations an vn ral is / / M M W W an vn ral is W W W A M / 47 W A 1 Ky paramtrs To mt th rquirmnt of FIR filtr, hoos Shap of th winow Duration of th winow Duration of th winow Trail an rror is not a satisfatory mtho to sign filtrs a simpl formalization of th winow mtho by Kaisr 48

25 Part III-C: Kaisr winow filtr sign Filtr sign IIR filtr sign Analog filtr sign IIR filtr sign by impuls invarian FIR filtr sign Commonly us winows Gnraliz linar-phas FIR filtr Th Kaisr winow filtr sign mtho 49 Th Kaisr winow filtr sign mtho An asy way to fin th tra-off btwn th mainlob with an si-lob ara Th Kaisr winow 1/ I 0[ 1 [ n / ] ], 0 n M w[ I 0 0, othrwis M / 50 I is th zroth-orr orr moifi Bssl funtion of 0 th first kin. 0is an austabl sign paramtr. Th lngth M+1 an th shap paramtr an b aust to tra si-lob amplitu for main-lob with not possibl for pring winows! 5

26 Th Kaisr winow 51 Dsign FIR filtr by th Kaisr winow Calulat M an to mt th filtr spifiation Th pak approximation rror is trmin by Dfin A 0log 10 thn Pak rror is fix for othr winows A 8.7, A A A 1, 1 A , A 1 Rtangular Passban utoff frquny is trmin by: 5 1 p Stopban utoff frquny s by: Transition with s p M must satisfy A 8 M.85 6

27 A lowpass filtr Spifiations Spifiations for a isrt-tim lowpass filtr , , 0.6 s p Dsign th lowpass filtr by Kaisr winow Dsigning by winow mtho iniating 1, w must st Transition with s p 0. A 0log10 60 Th two paramtrs: 5.653, M 37 Cutoff frquny of th ial lowpass filtr s p / 0. 5 Impuls rspons 54 sin n I h n 0, [ 1 [ n / ] 0 [ I 0 1/ ], 0 n M othrwis 7

28 Dsign th lowpass filtr What is th group lay? M/= Summary Filtr sign IIR filtr sign Analog filtr sign IIR filtr sign by impuls invarian FIR filtr sign Commonly us winows Gnraliz linar-phas FIR filtr Th Kaisr winow filtr sign mtho 56 8

29 Cours at a glan MM1 Disrt-tim signals an systms Systm MM Fourir transform an Z-transform Sampling an ronstrution Systm analysis DFT/FFT MM5 MM3 Filtr sign MM4 57 9

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