Digital Signal Processing. Hossein Mahvash Mohammadi

Transcription

1 Digitl Sigl Procssig Hossi Mhvsh Mohmmdi

2 Rfrcs Discrt-Tim Sigl Procssig b Opphim A.V., Schfr R.W, d Editio, Prtic-Hll, 999. Digitl Sigl Procssig with Computr Applictios, P. A. L, W. Furst Schum s Outli of Digitl Sigl Procssig, d Editio Schum's Outli Sris M.Hs

3 Grdig Assigmt % Proct % Midtrm Em %3 Fil Em %4 3

4 Cours Outlis Itroductio, Applictios, Histor Discrt-Tim sigls d Sstms Z Trsform Smplig Lir tim-ivrit sstm Filtr dsig Discrt d Fst Fourir Trsform 4

5 Itroductio Sic th ivtio of clculus i th 7th ctur, scitists d girs hv dvlopd modls to rprst phsicl phom i trms of fuctios of cotiuous vribls d diffrtil qutios. Mthmticis of th 8th ctur, such s Eulr, Broulli, d Lgrg, dvlopd mthods for umricl itgrtio d itrpoltio of fuctios of cotiuous vribl. 5

6 Itroductio Prior to 96s: Alog sigl procssig tcholog Th digitl computrs d microprocssors d dvlopmts of lgorithm such s th fst Fourir trsform FFT cusd mor shift to digitl tchologis. O of th first uss of DSP ws i oil prospctig, whr sismic dt ws rcordd o mgtic tp for ltr procssig. Cool d Tu 965 discovrd fficit lgorithm for computtio of Fourir trsforms FFT. Th ivtio of th microprocssor pvd th w for low-cost implmttios of discrt-tim sigl procssig sstms. 98s: IC tcholog improvs DSP, vr fst fid-poit d flotig-poit microcomputrs wr implmtd 6

7 DSP Applictio Img d Vido Applictios DVD, JPEG, Movi spcil ffcts, vido cofrcig, Mdicl MRI Mgtic Rsoc Imgig, Tomogrph, Elctrocrdiogrm, Militr Rdr, Sor, Spc photogrphs, rmot ssig, Mchicl Motor cotrol, procss cotrol, oil d mirl prospctig, 7

8 Discrt-tim sigls d sstms Sigl: somthig tht covs iformtio grll bout th stt or bhvior of phsicl sstm. Mthmticll rprstd s fuctio of idpdt vribls such s Tim: spch sigl Positio: digitl img Tim d positio: vido 8

9 Sigls Clssifictio Cotiuous-tim sigl Cotius-tim cotius mplitud : log sigl Empl: Spch sigl Cotius-tim discrt mplitud Empl: Trffic light Discrt-tim sigl Digitl Sigl: discrt mplitud => digitl sigl Empl : digitl img Discrt sigl: cotiuous mplitud Empl :smpls of log sigl, vrg mothl tmprtur 9

10 Discrt-tim sigls Sigl-procssig sstms Cotiuous-tim sstms: both th iput d th output r cotiuous-tim Discrt-tim sstms: both th iput d th output r discrt-tim sigls. Digitl sstm: both th iput d th output r digitl sigls. Discrt-tim sigls c b grtd b smplig cotiuous-tim sigl, or dirctl b som discrt-tim procss

11 Digitl vs Alog Pros Nois prformc Flibilit, usig grl computr Stbilit/duplicbilit Digitl storg, rdom ccss Cos Limittios of A/D & D/A Powr cosumptio

12 Discrt-Tim Sigls Squcs A Discrt-tim sigl is squcs of umbrs: ={ } - < < Priodic smplig of log sigl = T - < < T is th smplig priod f = / T smplig frquc log sigl digitl sigl digitl sigl A/D DSP D/A log sigl

13 Discrt-Tim Sigls Filtr is usd for ois rmovl d ti lisig Thr is lso low pss filtr ftr D/A log sigl digitl sigl digitl sigl A/D DSP D/A log sigl log sigl Low pss Filtr Smpl & Hold Qutizr Codig digitl sigl 3

14 Squc Oprtios Additio : = + w Multiplictio: = A. Shift dl = Product modultio =. w 4

15 Bsic Squcs Uit impuls Pls th sm rol s dlt fuctio Its dfiitio is simpl d prcis A squc c b prssd s: 5

16 Bsic Squcs Uit stp u... u u u u 6

17 Bsic Squcs A Epotil If A d α r rl umbrs, th th squc is rl. If < α < d A is positiv, th th squc vlus r positiv d dcrs with icrsig If α >, th th squc grows i mgitud s icrss. If < α <, ltrt i sig, but gi dcrs i mgitud with icrsig. 7

18 Bsic Squcs Siusoidl Acos A d φ r rl costt: 8

19 Compl umbr A compl umbr cosists of th rl d imgir prt: r im Mgitud : r im Phs : t im / r Polr form: cos si 9

20 Bsic Squcs Compl potil If α d A r compl : A A A, A A A A cos A si If α > potill growig vlop If α < potill dcig vlop If α = th squc is rfrrd to s compl potil squc A A cos A si

21 Bsic Squcs Compl potil squc A A cos A si Both th rl d imgir prts r siusoidl is th frquc d φ is th phs hs o uit th uit of must b rdi To hv closr log with th cotiuous-tim th uit of c b rdis/smpl th uit of is smpl

22 Bsic Squcs Compl potil squc A A cos A si With th frquc of ω +π: A A A Th sm rsults r obtid with ω +πr r ϵ Z Acos r Acos Thus th frquc is ol cosidrd i itrvl of: ω π or π< ω π

23 Bsic Squcs Compl potil priodicit: I th cotiuous tim siusoidl sigl is priodic with th priod of π/f I th discrt-tim : = + N for ll Acos Acos N ωn =π => N=π/ω d N must b itgr Th priodicit dpds o ω d it s ot cssril priodic with priod of π/ω Ep: ω =3π/4 => N= π / 3π/4 =8/3, => N=8, =3 ω = => N= π => thr is o itgr vlus for N d 3

24 Frquc domi i discrt-tim squcs Cotiuous- tim : t = A cosωt +ɸ s Ω icrss, t oscillts mor rpidl. Discrt- tim: = A cosω + ɸ s ω icrss from to π, oscillts mor rpidl. As ω icrss from π to π, oscillts slowr. Frqucis roud π r idistiguishbl from frqucis roud. Low frqucis rfr to frqucis roud π High frqucis rfr to ω roud π + π 4

25 Frquc domi i discrt-tim squcs 5

26 Discrt-Tim Sstm A trsformtio or oprtor tht mps iputs ito outputs squcs Empl = T{ } Idl dl sstm = d < < Movig vrg: Computs th vrg of M + M + smpls roud th th smpl M M M M 6

27 Discrt-Tim Sstm Mmorlss Sstms Th output t dpds ol o th iput t th sm vlu of Empl: Sstms with tim dl or tim dvc hv mmor d th sstm is ot mmorlss = d 7

28 Lir Sstms Lir sstms follow th pricipl of suprpositio: Additivit proprt T T T Homogit or sclig proprt T T Combitio of two proprtis Grl Form: T b T bt 8

29 Lir Sstms Empl: Accumultor => Lir 9

30 Nolir Sstm Empl : Empl : =, = => w=, w= = but w w => Nolir 3.. => Nolir log w

31 Tim-Ivrit Sstms A tim-ivrit or shift-ivrit sstm is sstm tht tim shift or dl of th iput squc cuss th sm shift i th output squc. Accumultor s Tim-Ivrit sstm: 3 Substitutig =, wh - < < => - - < <

32 Tim-Ivrit Sstms Th comprssor sstm Discrds M smpls out of M M M M M Othr solutio fidig coutrmpl: =δ, M=, =, = δ =δ = δ but ==δ- for = Th sstm is ot Tim-Ivrit 3

33 Cuslit I cusl sstm, th currt output of th sstm dpds ol o th currt d prvious iputs. If = for th = Empl: forwrd diffrc sstm: = + = δ, = ; = δ δ, = = for but for = Empl: bcwrd diffrc sstm = Th output dpds ol o th prst d pst vlus of th iput. So thr is o w for th output t spcific tim to icorport vlus of th iput for > o, th sstm is cusl. No-cusl 33

34 Stbilit A sstm is stbl if d ol if ch boudd iput squc producs boudd output squc. Th iput is boudd, wh thr is B tht: B < for ll I stbl sstm for vr boudd iput, th output must b boudd B < for ll Th proprtis dfid i this sctio r proprtis of sstms, ot of th iputs to sstm. It might tht th proprtis hold for som iputs, but th sstm dos ot hv th proprt bcus it dos ot hold th proprt for ll iputs. 34

35 Tstig for stbilit or istbilit Cosidr th sstm of : Assum tht th iput is boudd such tht: B for ll : B th B choosig B B it provs tht is boudd Th sstm log for vlus of th tht =. So th output is ot boudd d th sstm is ustbl., Accumultor: if = u => u Thr is o fiit choic for B such tht + B <, 35

36 Lir Tim-Ivrit Sstms Th sstms which r lir d tim-ivrit r LTI. LTI sstms r importt clss of sstms du to covit rprsttios d sigifict pplictios. A lir sstm c b compltl chrctrizd b its impuls rspos. Lt h b th rspos of th sstm to δ- Cosidrig th proprt of tim ivric: Th rspos of LTI sstm for iput c b obtid b th covolutio sum of iput with th impuls rspos h 36 h T T T } { } { } { h h

37 Output of LTI sstm 37

38 Computtio of th Covolutio Sum Clcultio of h-= h-- Rflctig h bout th origi to obti h- ; Shiftig th origi of th rflctd squc to =. 38

39 Alticl Evlutio of th Covolutio Sum Cosidr sstm with impuls rspos d iput: h u u N,, N othrwis u Th output t is th sums ovr ll of th product h For <, = bcus thr is o ovrlp btw d h 39

40 Alticl Evlutio of th Covolutio Sum If d N + or N : => For N 4 h N N N N N h, N N N N

41 Alticl Evlutio of th Covolutio Sum 4 N N N N othrwis N N u u h,, u

42 Proprtis of LTI Sstms b th Commuttiv Substitutio of m = m * h = h * Covolutio m h m Th sstm output is th sm if th rols of th iput d impuls rspos r rvrsd. Th impuls rspos of cscd combitio of lir tim-ivrit sstms is idpdt of thir ordr. Th covolutio oprtio lso distributs ovr dditio: m h m m h * 4

43 Proprtis of LTI Sstms b th Covolutio Cscd coctio Th output of th first sstm is th iput to th scod Th output of th scod is th iput to th third, tc. Th output of th lst sstm is th ovrll output. Th impuls rspos of two cscdd LTI sstms is th covolutio of th impuls rsposs of th two sstms. Th ovrll impuls rspos of th sstm is: h h * h 43

44 Proprtis of LTI Sstms b th Prlll coctio Covolutio th sstms hv th sm iput, d thir outputs r summd to produc ovrll output. Bsd o th distributiv proprt of covolutio th ovrll sstms is quivlt to sigl sstm whos impuls rspos is th sum of th idividul impuls rsposs *h + * h=*h+h 44

45 Proprtis of LTI Sstms b th Covolutio Stbilit Evr boudd iput producs boudd output. LTI sstms r stbl if d ol if th impuls rspos is bsolutl summbl: s h h h If is boudd: Substitutig B for : Thus, is boudd if s is B B h 45

46 Proprtis of LTI Sstms b th Covolutio Stbilit To show tht summblit of s is cssr it must b show tht if s =, th boudd iput c b foud tht will cus uboudd output. is boudd b uit: Th output t = : Thrfor, if s=, it is possibl for boudd iput squc to produc uboudd output squcs. 46,, * h h h h s h h h

47 Proprtis of LTI Sstms Rflctd i th Impuls Rspos Th impuls rsposs of th sstms c b computd b th rspos to δ: Idl dl = d < < h= δ-d Movig vrg M M M Forwrd diffrc =+ h=δ+-δ Bcwrd diffrc = h=δ-δ M h M M Accumultor, h,, M M othrwis 47

48 Proprtis of LTI Sstms b th Cuslit Covolutio I cusl sstms th output dpds ol o th iput smpls, for <. this dfiitio implis th coditio: h = for < h m m h m m h m m h * 48

49 Covrsio of ocusl to cusl sstm b dl Forwrd diffrc sstm cscdd with dl sstm is ocusl Th sm rsult is obtid if th squc is first dld d th comput forwrd diffrc h * * Th rsult is th sm s to th impuls rspos of th bcwrd diffrc A ocusl FIR sstm c b md cusl b cscdig it with sufficitl log dl. 49

50 Ivrs sstm A ccumultor is cscdd with bcwrd sstm h u * u u Th ovrll impuls rspos is th impuls h * h h * h i i Th output is qul to iput Th bcwrd diffrc sstm is th ivrs of ccumultor Ivrs sstms r usful wh it is cssr to compst th ffcts of lir sstm. 5

51 Lir Costt-Cofficit Diffrc Equtios LCCDE Th iput d th output stisf Nth-ordr lir costt-cofficit diffrc qutio of th form: Diffrc Equtio Rprsttio of th Accumultor Th output for : sprtig th trm from th sum Substitutio of qutios 5 N M m m m b

52 Diffrc Equtio Rprsttio of th Movig-Avrg Sstm th impuls rspos of th movig-vrg sstm with M= : W lso hv: Th impuls rspos c lso b rprstd s: 5 M u u m h M M * u M M h

53 Diffrc Equtio Rprsttio of th Movig-Avrg Sstm Diffrc qutio for th bloc digrm of th movig-vrg sstm : M M Th output of ccumultor sstm is: Th impuls rspos c lso b rprstd s: M M 53

54 Rcursiv Computtio of Diffrc Equtio If th iput d st of uilir vlus, for < is spcifid, th for c b dtrmid rcursivl b sprt =: for < - N c b clcultd b rrrgig th mi qutio ssumig tht th uilir coditios of -, -,..., - N r givsprt =N 54 N M m m m b N M b N M N N b N

55 Rcursiv Computtio of Diffrc Equtio Empl K - = c For > -: c 3... K c K c K 3 c c K K c K for 3 c 4 c 3 55

56 Rcursiv Computtio of Diffrc Equtio For < or 56 c c c c c for c ll for u K c

57 Lir Costt Cofficit Diffrc Equtio Solutio Dirct solutio cosists of two prts p is th prticulr solutio h is homogous solutio d it stisfis: h is th solutio of th qutio with = It c lso b solvd b z-trsform 57 h p N N M m m m b

58 Lir Costt Cofficit Diffrc Homogous solutio Equtio Solutio Th squc h is mmbr of fmil of solutios of th form: N h A m m Substitutig th bov squc ito w hv: N Thrfor, λm r roots of th bov qutio: m It is ssumd tht ll N roots of th bov polomil Equtio r distict. Sic h hs N udtrmid cofficits of, st of N uilir coditios is rquird for th uiqu spcifictio of for giv iput of N 58

59 Lir Costt Cofficit Diffrc Equtio Solutio Prticulr solutio p is i th scld form of th iput : Substitutig p d ito Fidig cofficits 59 p A p A M M M p m A... t cos A p t N M m m m b

60 Lir Costt Cofficit Diffrc Equtio Solutio Empl: Iput : Iitil coditios: Homogous prt: u,, 6 h c c 3 3, 3 6 6

61 Lir Costt Cofficit Diffrc Equtio Solutio Prticulr prt: Iput, costt: u, u p 3 p h c c

62 Lir Costt Cofficit Diffrc Equtio Solutio , 8 3,, 6 c c c c c c , c c c c c c, , c c 3 c c

63 Frquc-Domi Rprsttio Of Discrt-tim Sigls Ad Sstms Discrt-tim sigls m b rprstd i diffrt ws Compl potil squcs r igfuctios of lir tim-ivrit sstms th rspos to siusoidl iput is siusoidl with th sm frquc s th iput d with mplitud d phs dtrmid b th sstm. Th rprsttios of sigls i trms of siusoids or compl potils i.., Fourir rprsttios r vr usful i lir sstm thor. 63

64 Eigfuctios for Lir Tim- Ivrit Sstms Cosidr LTI sstm with th impuls rspos h d iput squc: Th corrspodig output is: If w dfi h Cosqutl, which is th sm s th iput is pprd i th output. It is clld igfuctio of th sstm, d th ssocitd igvlu is: H Th igvlu is clld th frquc rspos of th sstm d it dscribs th chg i compl mplitud of compl potil iput sigl s fuctio of th frquc ω. It cosists of th rl d imgir prts h H h H H H H H H R Im H 64

65 Frquc Rspos of th Idl Dl Sstm Cosidr th idl dl sstm: Cosidr iput of: Th output is: Th frquc rspos: Othr mthod b impuls rspos Th mgitud d phs r 65 d d d d H d h si cos d d d H d d H H,

66 Siusoidl Rspos of LTI Sstms Cosidr siusoidl iput: If h is rl: Thrfor Empl: Idl dl sstm 66 A A A cos A A H A A H H H A * H H cos A H H d H H, cos cos d d A A

67 Frquc Rspos of LTI sstms Th cocpt of th frquc rspos of LTI sstms is sstill th sm for cotiuous-tim d discrt-tim sstms. A importt diffrc is tht th frquc rspos of discrt-tim LTI sstms is priodic. r is itgr 67 h H H H r H H

68 Rprsttio of Squcs b Fourir Trsforms M squcs c b rprstd b Fourir itgrl of th form: is rprstd s suprpositio of ifiitsimll smll compl siusoids of th form: shows how much of ch frquc compot is rquird to sthsiz X X d X X 68

69 Impuls Rspos d Fourir Trsform Th frquc rspos of LTI sstm is th Fourir trsform of th impuls rspos. Th impuls rspos c b obtid from th ivrs Fourir trsform of frquc rspos 69 d H h h X H

70 Fourir Trsform d Stbilit To rprst squc b its IFT, must b boudd or i th rvrs rspct th ifiit sum must covrg. is th limit s M i th fiit sum: A sufficit coditio for covrgc c b foud s follows: If is bsolutl summbl th ist d thrfor ll stbl squcs hv Fourir trsforms. FIR sstms r stbl d thrfor th hv fiit, cotiuous frquc rspos. Wh squc hs ifiit lgth, w must b cocrd bout covrgc of th ifiit sum. 7 M M M X ll for X X X X X X X

71 Absolut Summbilit for Suddl- Applid Epotil Cosidr th followig squc d its Fourir trsform: X u Clrl, th coditio < is th coditio for th bsolut summbilit of : Absolut summbilit is sufficit coditio for th istc of Fourir trs- form rprsttio, d it lso gurts uiform covrgc. if if 7

72 Impuls Rspos of th Idl Lowpss Filtr Th frquc rspos of lowpss filtr H lp Th impuls rspos hip c b foud usig th Fourir trsform sthsis qutio: h ip sic hlp is ozro for <, th idl lowpss filtr is ocusl. Also, hlp is ot bsolutl summbl. Th squc vlus pproch zro s, but ol s /. This is bcus H lp is discotiuous t ω=ωc. c c c c d c c si c c c 7

73 Smmtr Proprtis of th Fourir Trsform A cougt smmtric squc: cougt ti-smmtric squc: A squc c b prssd s sum of cougt smmtric d cougt ti-smmtric squc: A rl squc tht is cougt smmtric is clld v squc rl squc tht is cougt tismmtric is clld odd squc. 73 * * o o o * * * * o o o o

74 Smmtr Proprtis of th Fourir Trsform A Fourir trsform c b dcomposd ito sum of cougt-smmtric d cougt tismmtric fuctios s B substitutig ω for ω i bov qutios, it follows tht is cougt smmtric d is cougt tismmtric: 74 o X X X * X X X * o X X X * o o X X * X X X X X o

75 Smmtr Proprtis of th Fourir Trsform 75

76 Fourir Trsform Thorms 76