Filter Design Techniques

Transcription

1 Fltr Dsgn chnqus Fltr Fltr s systm tht psss crtn frquncy componnts n totlly rcts ll othrs Stgs of th sgn fltr Spcfcton of th sr proprts of th systm ppromton of th spcfcton usng cusl scrt-tm systm Rlzton of th systm Fltr Dsgn-FIR cwlu@twns..nctu.u.tw 1

2 Rvw of scrttm systms Frquncy rspons : proc : pro for rl mpuls rspons h[] Mgntu rspons H s vn functon Phs rspons s o functon mpl : H yqust frquncy..., 1, 1,1, 1,1, Fltr Dsgn-FIR cwlu@twns..nctu.u.tw

3 Rvw of scrttm systms `Populr frquncy rsponss for fltr sgn : low-pss P hgh-pss HP bn-pss BP bn-stop mult-bn Fltr Dsgn-FIR cwlu@twns..nctu.u.tw 3

4 Rvw of scrttm systms FIR fltrs fnt mpuls rspons: H z B z 1 b + b z z + b z Movng vrg fltrs M fltrs pols t th orgn z hnc gurnt stblty zros zros of Bz, ll zro fltrs corrspons to ffrnc quton y[ ] b u [ ] + b1u [ 1] b u [ mpuls rspons h[ ] b, h[1] b1,..., h[ ] b, h[ 1] + ],... Fltr Dsgn-FIR cwlu@twns..nctu.u.tw 4

5 nr Phs FIR Fltrs on-cusl zro-phs fltrs : mpl: symmtrc mpuls rspons h[-],.h[-1],h[],h[1],...,h[] h[]h[-], 1.. frquncy rspons s H +. h[ ]... cos -.. rl-vlu zro-phs trnsfr functon - cusl mplmntton by ntroucng group ly Fltr Dsgn-FIR cwlu@twns..nctu.u.tw 5

6 nr Phs FIR Fltrs Cusl lnr-phs fltrs non-cusl zro-phs + ly mpl: symmtrc mpuls rspons & vn h[],h[1],.,h[] vn h[]h[-],.. frquncy rspons s H. h[ ]... cos.. cusl mplmntton of zro-phs fltr, by ntroucng group ly z z Fltr Dsgn-FIR cwlu@twns..nctu.u.tw 6

7 nr Phs FIR Fltrs yp-1 yp- yp-3 yp-4 vn +1o vn +1o symmtrc symmtrc nt-symmtrc nt-symmtrc h[]h[-] h[]h[-] h[]-h[-] h[]-h[-] / cos / cos cos / sn 1 cos /. sn cos P/HP/BP P/BP zro t zro t, zro t HP Fltr Dsgn-FIR cwlu@twns..nctu.u.tw 7

8 nr Phs FIR Fltrs ffcnt rct-form rlzton. mpl: u[] + Δ Δ Δ Δ Δ Δ Δ Δ bo b1 b b3 b4 y[] PS: IIR fltrs cn EVER hv lnr-phs proprty! Fltr Dsgn-FIR cwlu@twns..nctu.u.tw 8

9 Fltr Spcfcton E: ow-pss 1. 1 δ P 1.8 Pssbn Rppl P S.6 Pssbn Cutoff -> <- Stopbn Cutoff.4. Stopbn Rppl δ S Fltr Dsgn-FIR cwlu@twns..nctu.u.tw 9

10 Fltr Dsgn Problm Dsgn of fltrs s problm of functon ppromton For FIR fltr, t mpls polynoml ppromton For IIR fltr, t mpls ppromton by rtonl functon of z Fltr Dsgn-FIR cwlu@twns..nctu.u.tw 1

11 Fltr Dsgn-FIR 11 Fltr Dsgn by Optmzton I Wght st Squrs Dsgn : slct on of th bsc forms tht yl lnr phs.g. yp-1 spcfy sr frquncy rspons P,HP,BP, optmzton crtron s whr s wghtng functon cos / / H / H ,...,,...,,..., mn mn F W H H W + + W

12 Fltr Dsgn-FIR 1 Fltr Dsgn by Optmzton ths s quvlnt to stnr Qurtc Optmzton problm [ ] [ ]... cos... cos 1... } mn{ 1,..., + μ μ c c W p c c W Q p Q F p Q OP 1

13 Fltr Dsgn by Optmzton Empl: ow-pss sgn 1. Pssbn Rppl 1 1, <, S P pss - bn stop - bn optmzton functon s Pssbn Cutoff -> <- Stopbn Cutoff Stopbn Rppl F.. P ,..., 1 + γ.... W pss - bn S stop - bn Fltr Dsgn-FIR cwlu@twns..nctu.u.tw 13

14 Fltr Dsgn-FIR 14 Fltr Dsgn by Optmzton smplr problm s obtn by rplcng th F.. by whr th w sr st of n smpl frquncs h qurtc optmzton problm s thn quvlnt to lst-squrs problm +++ : smpl --- : unprctbl bhvor n btwn smpl frquncs. c W W F :,..., { { { } mn{ mn b b b b c c W b S 1 Compr to p.1

15 Fltr Dsgn by Optmzton thn ll ths s oftn supplmnt wth tonl constrnts Empl: ow-pss P sgn contnu pss-bn rppl control : 1 δ P, < P δ P s pss - bn rppl stop-bn rppl control : δs, S δs s stop - bn rppl Fltr Dsgn-FIR cwlu@twns..nctu.u.tw 15

16 Fltr Dsgn by Optmzton Empl: ow-pss P sgn contnu rlstc wy to mplmnt ths constrnts, s to mpos th constrnts only on st of smpl frquncs, P1 P,..., Pm n th pss-bn n, S1 S,..., Sn n th stop-bn h rsultng optmzton problm s : mnmz : F,..., [... ] subct to pss-bn constrnts P bp stop-bn constrnts S bs `Qurtc nr Progrmmng problm... 1 Fltr Dsgn-FIR cwlu@twns..nctu.u.tw 16

17 Fltr Dsgn-FIR 17 Fltr Dsgn by Optmzton II `Mnm Dsgn : slct on of th bsc forms tht yl lnr phs.g. yp-1 spcfy sr frquncy rspons P,HP,BP, optmzton crtron s whr s wghtng functon ]cos [ / / H / H m mn m mn,...,,..., W H H W W

18 Fltr Dsgn by Optmzton Concluson: I wght lst squrs sgn II mnm sgn prov gnrl `frmwor, procurs to trnslt fltr sgn problms nto stnr optmzton problms In prctc n n ttboos: mphss on spcfc -hoc procurs : - fltr sgn bs on wnows - qu-rppl sgn Fltr Dsgn-FIR cwlu@twns..nctu.u.tw 18

19 Fltr Dsgn-FIR 19 Fltr Dsgn usng Wnows Empl : ow-pss fltr sgn l low-pss fltr s hnc l tm-omn mpuls rspons s trunct h [] to +1 smpls : group ly to turn nto cusl fltr < < < 1 C C H H h c c α sn... 1 ] [. + < < othrws / / ] [ ] [ h h on-cusl n nfntly long

20 Fltr Dsgn usng Wnows Empl : ow-pss fltr sgn contnu not : t cn b shown tht tm-omn truncton corrspons to solvng wght lst-squrs optmzton problm wth th gvn H, n wghtng functon truncton corrspons to pplyng rctngulr wnow : h[ ] h [ ] w[ ] w[ ] 1 W 1 / < < othrws +++: smpl procur lso for HP,BP, --- : truncton n th tm-omn rsults n Gbbs ffct n th frquncy omn,.. lrg rppl n pss-bn n stop-bn, whch cnnot b ruc by ncrsng th fltr orr. / Fltr Dsgn-FIR cwlu@twns..nctu.u.tw

21 Fltr Dsgn usng Wnows Rmy : pply wnows othr thn rctngulr wnow: tm-omn multplcton wth wnow functon w[] corrspons to frquncy omn convoluton wth Wz : h[ ] h [ ] w[ ] H z H z W z cnt wnows : Hn, Hmmng, Blcmn, Ksr,. s ttboos wnow choc/sgn tr-off btwn s-lob lvls fn p pss-/stop-bn rppl n wth mn-lob fns trnston bnwth Fltr Dsgn-FIR cwlu@twns..nctu.u.tw 1

22 Wnowng Effct Gbbs phnomnon Fltr Dsgn-FIR

23 Wnowng 3

24 Equrppl Dsgn Strtng pont s mnm crtron,.g. mn mw mn,...,,..., m E Bs on thory of Chbyshv ppromton n th ltrnton thorm, whch roughly stts tht th optml sr such tht th m mmum wght ppromton rror s obtn t + trml frquncs m E E for 1,.., + tht hnc wll hbt th sm mmum rppl qurppl Itrtv procur for computng trml frquncs, tc. Rmz chng lgorthm, Prs-McCllln lgorthm Vry flbl, tc., vlbl n mny softwr pcgs Dtls omtt hr s ttboos Fltr Dsgn-FIR cwlu@twns..nctu.u.tw 4

25 Softwr FIR Fltr sgn bunntly vlbl n commrcl softwr Mtlb: bfr1n,wn,typ,wnow, wnow lnr-phs FIR sgn, n s fltr orr, Wn fns bn-gs, typ s `hgh,`stop, bfrn,f,m,wnow, wnow FIR sgn bs on nvrs Fourr trnsform wth frquncy ponts f n corrsponng mgntu rspons m brmzn,f,m, qurppl lnr-phs FIR sgn wth Prs-McCllln Rmz chng lgorthm Fltr Dsgn-FIR cwlu@twns..nctu.u.tw 5