IV. The z-transform and realization of digital filters

Transcription

1 Digitl Sigl Procssig 4 Dcmbr 6, 9 IV. Th -trsform d rlitio of digitl filtrs 7 Syllbus: Rviw of -trsforms, Applictios of -trsforms, Solutio of diffrc qutios of digitl filtrs, Bloc digrm rprsttio of lir costt-cofficit diffrc qutios, Bsic structurs of IIR systms, Trsposd forms, Bsic structurs of FIR systms, Systm fuctio. Cotts: 4. Itroductio 4. Importt proprtis of -trsforms 4. Trsforms of som usful squcs 4.4 Rgio of covrgc d stbility 4.5 Ivrs -trsform by prtil frctios 4.6 Rltioships mog systm rprsttios 4.7 Ivrs -trsform by powr sris xpsio log divisio 4.8 Computtio of frqucy rspos 4.9 Z-trsforms with iitil coditios 4. Stdy-stt d trsit rsposs for first ordr systm 4. Rlitio of digitl filtrs 4. Th Lttic structur Itroductio 4. *Ivrs -trsform by complx ivrsio itgrl DSP-4 Z of 84 Dr. Rvi Bill

2 Itroductio For cotiuous-tim systms th Lplc trsform is xtsio of th Fourir trsform. Th Lplc trsform c b pplid to brodr clss of sigls th th Fourir trsform c, sic thr r my sigls for which th Fourir trsform dos ot covrg but th Lplc trsform dos. Th Lplc trsform llows us, for xmpl, to prform trsform lysis of ustbl systms d to dvlop dditiol isights d tools for LTI systm lysis. Th -trsform is th discrt-tim coutrprt of th Lplc trsform. Th - trsform bls us to ly crti discrt-tim sigls tht do ot hv discrt-tim Fourir trsform. Th motivtios d proprtis of th -trsform closly rsmbl thos of th Lplc trsform. Howvr, s with th rltioship of th cotiuous tim vrsus th discrt-tim Fourir trsforms, thr r distictios btw th Lplc trsform d th - trsform. Dfiitio Th two-sidd biltrl -trsform, X, of th squc x is dfid s X ʓ{x} x whr r is th complx vribl. Th bov powr sris is Lurt sris. Th o sidd uiltrl -trsform is dfid s X + x Th uiltrl -trsform is prticulrly usful i lyig cusl systms spcifid by lir costt-cofficit diffrc qutios with oro iitil coditios ito which iputs r stppd. It is xtsivly usd i digitl cotrol systms. Im -pl ROC R R x R x+ Th rgio of covrgc ROC is th st of vlus for which th bov summtio covrgs. I grl th ROC is ulr rgio i th complx -pl giv by ROC R x < < R x+ Rltioship btw th -trsform d th discrt-tim Fourir trsform Sttig r i th dfiitio givs us X x r [ r x ] r DSP-4 Z of 84 Dr. Rvi Bill

3 If r, th th -trsform, vlutd o th uit circl, givs th discrt-tim Fourir trsform of th squc x, i.., X x X Exmpl 4.. Th positiv-tim sigl xt t, t, othrwis is smpld t T-scod itrvls rsultig i th squc xt or x x t t T T T, T d x,, < If < this squc dcys xpotilly to s. Substitutig x ito th dfiig qutio, th -trsform is ʓ{x} X, <, > -pl Im ROC, > Shdd r Pol t R Zro t Th ROC is >. This X is rtiol fuctio rtio of polyomils i. Th roots of th umrtor polyomil r th ros of X d th roots of th domitor polyomil r th pols of X. This is right-sidd squc. Right-sidd squcs hv ROC tht is th xtrior of circl with rdius R x > i this cs. If th ROC is th xtrior of circl it is rightsidd squc. Dfiitio A right-sidd squc x is o for which x for ll < whr is positiv or gtiv but fiit. If th x is cusl or positiv-tim squc. DSP-4 Z of 84 Dr. Rvi Bill

4 Exmpl 4.. Th gtiv-tim squc x b u. Rcll tht th uit stp squc u. if th rgumt of u. is, i.., if or. x b,, othrwis If b > this squc dcys xpotilly to s. Th -trsform is, ʓ{x} X x b Lt m d chg th limits ccordigly to gt, X m m b m b m b W ddd i th lst stp bov to m up for th m trm withi th summtio. Th rsult is, X, b < b, ROC is < b b Im -pl ROC b pol R ro This is lft-sidd squc. Such squc hs rgio of covrgc which is th itrior of circl, < R x+. I this cs th ROC is < b. ot tht if b th th two xmpls bov hv xctly th sm X. So wht ms th diffrc? Th rgio of covrgc ms th diffrc. Dfiitio A lft-sidd squc x is o for which x for ll, whr is positiv or gtiv but fiit. If th x is ticusl or gtiv-tim squc. DSP-4 Z 4 of 84 Dr. Rvi Bill

5 Exmpl 4.. [Two-sidd squc] This is th sum of th positiv- d gtiv-tim squcs of th prvious two xmpls. y, u b u b, < Substitutig ito th dfiig qutio, Y ʓ{y} ow, from Exmpls d, [ u b u ] ROC > & b b ROC < b b So, th dsird trsform Y hs rgio of covrgc qul to th itrsctio of th two sprt ROC s > d < b. Thus Y, with ROC { > } { < b } b b, with ROC < < b b Th ROC is th ovrlp of th shdd rgios, tht is, th ulr rgio btw d b. Th two ros r t d +b/, d th two pols t d b. Wh < b Im ro ro +b/ b R ROC pols DSP-4 Z 5 of 84 Dr. Rvi Bill

6 If b < th trsform dos ot covrg. Im ro > ro b R pols I th bov thr xmpls w my xprss th -trsform both s rtio of polyomils i i.., positiv powrs d s rtio of polyomils i gtiv powrs. From th dfiitio of th -trsform, w s tht for squcs which r ro for <, X ivolvs oly gtiv powrs of. Howvr, rfrc to th pols d ros is lwys i trms of th roots of th umrtor d domitor xprssd s polyomils i. Also, it is somtims covit to rfr to X, writt s rtio of polyomils i i.., positiv powr of, s hvig pols t ifiity if th dgr of th umrtor xcds th dgr of th domitor or ros t ifiity if th umrtor is of smllr dgr th th domitor. Exmpl 4..4 [Fiit-lgth squc] Oly fiit umbr of squc vlus r o-ro, s giv blow. x for < d for >, whr d r fiit o-ro for By th dfiig qutio w hv X x x x Covrgc of this xprssio rquirs simply tht x < for. Th my t o ll vlus xcpt if is gtiv d if is positiv. Thus th ROC is t lst < < d it my iclud ithr or dpdig o th sig of d. 4. Importt proprtis of -trsforms Th proofs r sily obtid by usig th bsic -trsform dfiitio d trsformtios i th summtio. [Sc. Opphim & S] DSP-4 Z 6 of 84 Dr. Rvi Bill

7 Lirity If ʓ[x] X with ROC r x < < r x d ʓ[y] Y with ROC r y < < r y th ʓ[ x + b y] X + b Y with ROC t lst th ovrlp of th ROC s of X d Y. If thr is y pol-ro cclltio du to th lir combitio, th th ROC my b lrgr. Trsltio Tim-shiftig If ʓ[x] X with ROC r < < r th ʓ[x ] X with th sm ROC xcpt for th possibl dditio or dltio of or du to. Exmpl Giv x {, } d x x+ fid X d X d thir rspctiv ROCs. X +, ROC: tir -pl xcpt ; X +, ROC: tir - pl xcpt. ultiplictio by complx xpotil squc Sclig i th -domi If ʓ[x] X with ROC r < < r th ʓ[ x] X with ROC r / < < r. Exmpl Giv x {, } d x.5 x fid X d X d thir rspctiv ROCs. 4 ultiplictio by rmp If ʓ[x] X with ROC r < < r th dx ʓ[ x] with ROC r < < r. d Exmpl Giv x {, } d x + + x fid X d X d thir rspctiv ROCs. ʓ[x ] ʓ[] + ʓ[ x] + ʓ[ [ x]] 5 Tim rvrsl If ʓ[x] X with ROC r < < r th ʓ[ x ] X with ROC / / r r < < Exmpl Giv x u d X X dtrmi x. x.5 u, X, ROC:.5 <.5 ʓ - { X } x ; x ʓ - { X } x +. u 6 Covolutio i tim domi lds to multiplictio i frqucy domi Giv ʓ[x] X with ROC R x d ʓ[y] Y with ROC R y d x*y x h th ʓ[x*y] X.Y with ROC R x R y. 7 ultiplictio i tim domi lds to covolutio i frqucy domi If ʓ[x] X with ROC r x < < r x d ʓ[y] Y with ROC r y < < r y th ʓ[x.y] X v Y v dv, ROC r x r y < < r x r y. v C whr is complx cotour itgrl d C is closd cotour i th itrsctio of th ROCs C of Xv d Y/v. DSP-4 Z 7 of 84 Dr. Rvi Bill

8 8 Iitil Vlu Thorm If x is cusl squc with trsform X, th x lim X 9 Fil Vlu Thorm If ʓ[x] X d th pols of X r ll isid th uit circl th th vlu of x s is giv by x lim[ X ] Som lso giv this s x lim[ X ] 4. Trsforms of som usful squcs Th uit smpl δ: ʓ[δ]., ROC ll Dlyd uit smpl δ : ʓ[δ ]., ROC > if > < if < Uit stp u positiv tim: ʓ[u] u + + +, ROC < or > Uit stp u gtiv tim: ʓ[ u ] u + + +, ROC < 4 Expotil u, drivd i rlir xmpl: ʓ[ u ], ROC > 5 Expotil b u ; gtiv tim; drivd rlir: ʓ[ b u ], ROC < b b 6 Uit rmp u. Giv tht ʓ[u] U du d [..] ʓ[ u] d d ROC >, sm s tht of U 7 Siusoid si u : ʓsi u si, ROC > cos DSP-4 Z 8 of 84 Dr. Rvi Bill

9 DSP-4 Z 9 of 84 Dr. Rvi Bill ʓ si u si, ROC < or > Usig th idtitis cos d si w hv ʓ si u cos si, ROC > As xtsio, usig proprty #, ʓ si u cos / / si / cos si, ROC > 8 Cosiusoid cos u. Usig th rltio cos d procdur similr to tht for th siusoid w gt ʓ cos u cos cos, ROC > As xtsio, usig proprty #, ʓ cos u cos / / cos / / cos cos, ROC > 4.4 Rgio of covrgc d stbility Suppos x is cusl squc tht c b writt s sum of complx xpotils. This ts i wid clss of sigls icludig siusoids, xpotils, d products throf. Lt x i i u Tig th trsform of x givs ʓ[x] X i i

10 Th rgio of covrgc R is th itrsctio of th rgios of covrgc for ch xpotil s follows: R R i whr R i {: > i } i Thrfor, R {: > lrgst of i } s show hr Figur -pl Im ROC, > Lrgst i Lrgst i R All othr i isid circl Sic th ROC for trsltd xpotil rmis th sm s tht for th origil xpotil, ll right-sidd squcs tht r sums of trsltd xpotils hv ROCs similr to tht xprssd bov. By similr rgumt ll lft-sidd squcs xprssibl s sum of trsltd complx xpotils hv ROC, L, giv by L {: < smllst of b i } If w hv combitio of right- d lft-sidd squcs, th corrspodig ROC is th itrsctio of R d L. Thrfor th totl ROC bcoms ulr rgio s show blow d giv by R Totl R L {: Lrgst of i < < smllst of b i } Aulr rgio is th totl ROC Im Lrgst i Smllst b i R All othr i isid th ir circl All othr b i outsid th outr circl DSP-4 Z of 84 Dr. Rvi Bill

11 Th stbility of systm with impuls rspos tht is th sum of trsltd right- d lft-sidd squcs c b dtrmid from th rgio of covrgc. Assum tht h is th uit smpl rspos of cusl or o-cusl lir shift-ivrit systm. Lt ʓ[h] H, th so-clld systm fuctio. Th: Thorm A lir shift-ivrit systm with systm fuctio H is BIBO stbl if d oly if th ROC for H cotis th uit circl. This thorm c b usd to dtrmi stbility for giv H without obtiig th impuls rspos or chcig outputs for ll boudd iput sigls. Illustrtio of stbility d cuslity For A systm fuctio with pols t, sy,.5, d.5, thr r thr possibl rgios of covrgc. ROC is.5 < <.5. Hr th systm is stbl sic th uit circl is isid th rgio of covrgc. Th impuls rspos, h, is two-sidd, so th systm is ocusl. Im ROC Uit circl.5.5 R DSP-4 Z of 84 Dr. Rvi Bill

12 ROC is <.5. Hr th systm is ot stbl. Th impuls rspos, h, is lftsidd, so th systm is ocusl. Im ROC Uit circl.5.5 R ROC is >.5. Hr th systm is ot stbl. Th impuls rspos, h, is rightsidd, so th systm my b cusl. Im ROC.5.5 R 4.5 Ivrs -trsform by prtil frctios Asid Compriso of ivrs -trsform mthods A limittio of th powr sris mthod is tht it dos ot ld to closd form solutio lthough this c b dducd i simpl css, but it is simpl d lds itslf to computr implmttio. Howvr, bcus of its rcursiv tur cr should b t to miimi possibl build-up of umricl rrors wh th umbr of dt poits i th ivrs -trsform is lrg, for xmpl by usig doubl prcisio. DSP-4 Z of 84 Dr. Rvi Bill

13 Both th prtil frctio mthod d th ivrsio itgrl mthod rquir th vlutio of rsidus lbit prformd i diffrt wys. Th prtil frctio mthod rquirs X th vlutio of th rsidus of X or. Th complx ivrsio itgrl rquirs th vlutio of th rsidus of X. I my istcs vlutio of th complx ivrsio itgrl is dlssly difficult d ivolvd. Both th prtil frctio mthod d th ivrsio itgrl mthod ld to closd form solutios. Th mi disdvtg is hvig to fctori th domitor polyomil of X wh it is of ordr grtr th. Aothr disdvtg is multipl ordr pols d th rsultig diffrtitios wh dtrmiig rsidus. Th prtil frctio mthod dirctly grts th cofficits of prlll structurs for digitl filtrs. Th ivrsio itgrl mthod is widly usd i th lysis of qutitio rrors i discrt-tim systms. Ed of Asid As i Lplc trsforms, i ordr to xpd rtiol fuctio ito prtil frctios, th dgr of th umrtor should b lss th th dgr of th domitor propr frctio. If it is ot th w prform log divisio s blow whr Q is th quotit d is th rmidr. X Q + D D Log Divisio Q Quotit Domitor D umrtor --- Rmidr Th log divisio is do util w gt rmidr polyomil whos dgr is lss th th dgr of th domitor D. W th obti x s x ʓ - {X} ʓ - {Q} + ʓ - D Sic /D is propr frctio it c b xpdd ito prtil frctios. Th ovrll ivrs trsform is obtid by looig up tbl of -trsform pirs. Howvr, thr is ltrtiv vilbl i th cs of -trsforms which is ot vilbl i Lplc trsforms. This is rsult of th fct tht -trsforms r chrctrid by i th umrtor s c b vrifid by looig t tbl of -trsforms. Thrfor, istd of xpdig X w my, istd, xpd [X/] ito prtil frctios givig X A B + + so tht X is giv by A B X + + This c b ivrtd by simpl loo-up of tbl of trsforms. ot lso tht i som css X /D my ot b propr frctio but [X/] is d, thrfor, this mthod obvits th d for log divisio of by D. I still othr css v [X/] my ot b propr frctio. S ltr udr Grl procdur for prtil frctio xpsio. DSP-4 Z of 84 Dr. Rvi Bill

14 Exmpl 4.5. S lso log divisio ltr. Fid th ivrs -trsform, usig prtil frctios, of X D This is ot propr frctio sic th dgr of th umrtor is ot lss th th dgr of th domitor. Howvr, X/ is propr frctio X which hs th prtil frctio xpsio X + or X + By looig up tbl of -trsforms th ivrs -trsform is ʓ - {X} x u + u ot tht w r givig hr th cusl solutio tht corrspods to ROC > ot < < or < so tht x is right-sidd squc. Th ltrtiv mthod is to divid by D s blow s is stdrd prctic i Lplc trsforms. ot tht i this log divisio th umrtor d domitor polyomils r rrgd i th ordr of dcrsig powrs of. Thr r thr othr wys ll of thm wrog of rrgig th two polyomils for th log divisio. Log Divisio Quotit Domitor + umrtor Rmidr Thus X c b xprssd s 4 4 X + +X whr X X is propr frctio d c b xpdd ito prtil frctios s blow: 4 A B X + Solvig for A d B w gt A d B, so tht X my b writt X + + Tig th ivrs -trsform w gt x ʓ - {X} ʓ - ʓ - {} + ʓ - + ʓ - A trm li ʓ -. is hdld by writig s. W ow tht ʓ - u, so ʓ - u DSP-4 Z 4 of 84 Dr. Rvi Bill

15 Similrly ʓ -. u. Thus x δ + u +. u This c b vrifid to b quivlt to x u + u obtid rlir. I ATLAB Prtil frctios Th prtil frctios my b computd by usig th rsidu fuctio. I this mthod X is rrgd s rtio of polyomils i gtiv powrs of d, i th domitor, th ldig cofficit. S Grl procdur for prtil frctio xpsio ltr. R R X K + + p p W dfi th cofficit vctors b [, -] d [, -, ]; R [R, R ] rprsts th rsidus prtil frctio costts, p [p, p ] th pols d K costt. %Prtil frctios b [, -], [, -, ], [R, p, K] rsidu b, Th ATLAB rsults rturd r R p K [] Th ATLAB output tlls us tht th pols r t d d th corrspodig rsidus r, rspctivly, d. Furthr K. Thrfor, X X ot tht th X + obtid by th rsidu fuctio d + r th sm sic X hs o rptd pols. This wo t b th cs if X hs rptd pols. Exmpl 4.5. [] Fid if th discrt LTI systm dscribd by y y +.5 y x + x is BIBO stbl or ot. Fid its trsfr fuctio d impuls rspos. Stch its pol-ro plot. Solutio T th -trsform of both sids: ʓ{y y +.5 y } ʓ{x + x } Y Y +.5 Y X + X Y +.5 X + DSP-4 Z 5 of 84 Dr. Rvi Bill

16 H Y X.5 /.5 /.5 Th domitor hs roots t 4...5, d.5.5 Thus th trsfr fuctio H hs ros t,, d pols t.5 ±.5. For cusl systm right-sidd squc, h th rgio of covrgc is > or >.77. Figur Im Uit circl Pol t R Zro t Zro t Pol t.5.5 Th impuls rspos is giv by h ʓ - {H}. W d prtil frctios for H; w shll istd hdl H/: H A Solvig for A d A *, w gt A * A t / A * Thus w hv * H A A H A + A * b DSP-4 Z 6 of 84 Dr. Rvi Bill

17 whr t / 4 b.5.5 Th ivrs -trsform is 4 / So tht for, h A + A * b,, othrwis h A / 4 A / 4 + A * / 4 + A * / 4 / 4 + / 4 + / 4 / 4 / / 4 / 4 + / 4 / 4 / 4 cos /4 cos /4 + / 4 / 4 si /4 To sum up, h [cos /4 + si /4],, othrwis Altrtivly, sic th two trms i / 4 h A + A * / 4, r complx cougts of ch othr w c writ / 4 h R A, DSP-4 Z 7 of 84 Dr. Rvi Bill

18 Altrtiv I th bov solutio th impuls rspos iitilly cotis complx umbrs; ths hv b lgbriclly mipultd ito si d cosi trms. A mor dirct wy to obti th impuls rspos i form tht cotis o complx umbrs is to us rsults #7 d #8 i Trsforms of som usful squcs d mipult th trsform H.5.5 ito thos forms. Comprig th domitor of H with th domitor of th trsforms of th si d cosi fuctios ʓ si u si, ROC > cos d ʓ cos u cos, ROC > cos w gt.5 cos from which.5, cos ω /4, cos, cos ½, si, si ½ Th umrtors of th two trsforms th r si d cos I light of ths w mipult th umrtor of H so tht it will coti d umrtor Domitor. 5 Thus H W hv, i ffct, rrgd H s H + Thrfor, h ʓ - {H} ʓ - + ʓ DSP-4 Z 8 of 84 Dr. Rvi Bill

19 Imgiry Prt cos / 4 u + si / 4 u I ATLAB Pol-ro plot It is covit to spcify th trsfr fuctio s rtio of polyomils i H.5 Th umrtor cofficits, from lft to right, r {b i, i to }, spcifyig th vctor b [b, b ] [, ]. Similrly, th domitor cofficits r { i, i to } from lft to right, with spcifyig th vctor [,, ] [, -,.5]. %Pol-ro plot b [, ]; [, -,.5]; pl b, Rl Prt I ATLAB Prtil frctios Th prtil frctios my b computd by usig th rsidu fuctio s blow. ot tht H is rrgd s rtio of polyomils i gtiv powrs of. R R H K p p W dfi th cofficit vctors b [, ] d [, -,.5]; R [R, R ] rprsts th rsidus prtil frctio costts, p [p, p ] th pols d K costt. %Prtil frctios b [, ], [, -,.5], DSP-4 Z 9 of 84 Dr. Rvi Bill

20 [R, p, K] rsidu b, Th ATLAB rsults rturd r R.5 -.5i.5 +.5i p.5 +.5i.5 -.5i K [] Thrfor,.5.5 H Exmpl 4.5. Fid th ivrs trsform of X, whr th ROC is >, 4 b < /, c / < <. Solutio Th thr possibl rgios of covrgc r show blow. Th xmpl shows tht th ivrs trsform, x, is uiqu oly wh th ROC is spcifid. b c For ROC > X 4 A B + / / A / /, d B / DSP-4 Z of 84 Dr. Rvi Bill / X / / + / / / X + / Th ivrs is / / x ʓ - {X} ʓ - / ʓ - / / + ʓ - /

21 Th ROC is outsid th lrgst pol sigifyig right-sidd squc for ch pol. Th ivrs bcoms x u u u u b For ROC < /. Th prtil frctio xpsio dos ot chg. Sic th ROC is iwrd of th smllst pol, x cosists of two gtiv-tim squcs. x ʓ - {X} ʓ - / / + ʓ - / u / u / / u c For ROC / < <. Th prtil frctio xpsio stys th sm. Th pol t corrspods to gtiv-tim squc lft-sidd squc whil th pol t / givs positiv-tim squc right-sidd squc. / / x ʓ - {X} ʓ - / ʓ - / / + ʓ - / u Th ovrll rsult is two-sidd squc. DSP-4 Z of 84 Dr. Rvi Bill u u Exmpl Somtims thr is o i th umrtor to fctor out, but w still c divid X by s i this xmpl. Fid x for X whr th ROC is >. 4 Solutio X A B C / / A B C / / / X + / X + / x δ + u / u Exmpl [] Fid th ivrs -trsform of X...56 Solutio Th roots of th qudrtic i th domitor r giv by u

22 b b 4c...4 X.8.7 X A B C....5 A B C / / d.7 X.79 / /.75 X x ʓ - {X}.79 δ +.8 u u Exmpl Fid th ivrs -trsform of X / / 4 Solutio X / / 4 A / / 4 B C / 4 / / / A B C + + / / 4 8 X / / X 8 + / / 4 x 8 δ + 8 u 6 u 4, > ½. DSP-4 Z of 84 Dr. Rvi Bill

23 4 Exmpl Fid th ivrs of X / / 4 DSP-4 Z of 84 Dr. Rvi Bill for ROC ½ < < X / / 4 / 4 / 8 Thr is pol t. Th umrtor dgr is d is grtr th th domitor dgr. By log divisio w rduc th umrtor dgr by so tht th rsultig umrtor dgr is lss th tht of th domitor dgr. X /6 / + / 4 / 8 4 / 4 / 8 ot tht i th log divisio ldig to th bov rsult th umrtor d domitor polyomils r rrgd i th ordr of dcrsig powrs of. Thr r thr othr wys ll of thm wrog of rrgig th two polyomils for th log divisio. Th propr frctio prt c ow b xpdd ito prtil frctios: /6 / A B + / 4 / 8 / / 4 A B X X /6 / / 4 /6 / / 4 4 / / 4 + / 5/ 7/6 5 / 7 /6 + / 4 5 / 7 /6 + / / x δ+ + + u Th swr hs vlus for d du to th pol t. Th rsultig x is ot cusl squc. I ATLAB Prtil frctios Th trsform X rprsts ocusl squc. 4 X 4 / 4 /8 / 4 /8 Prtil frctios cot b computd by usig th rsidu fuctio dirctly o X sic. Howvr, X /6 / + 4 +X / 4 /8 4 whr /6 / X /6 / / 4 /8 / 4 /8 O which w my us th rsidu fuctio. Exmpl Assumig tht H idpdtly of ch othr 4 5 is cusl systm fuctio, prov th followig

24 Solutio h 4/5 + 9/5 b h 5 u + 4 h + 9 H 4 / 5 9/ u u- u-, b H , c By log divisio H Exmpl Prtil frctios c b obtid with th -trsform, sy H, xprssd s rtio of polyomils i gtiv powrs of. This mouts to xpdig H/ ito prtil frctios. Hr is xmpl: 8 4 H / 4 / This xmpl is from Prlll rlitio of IIR filtrs, towrds th d of this Uit whr w obti H / / Howvr, w my lso procd with gtiv powrs of s blow w my viw p s w vribl: / 4 /.5.5 H By log divisio w rduc th dgr of th umrtor by d th xpd th propr frctio prt ito prtil frctios: Log Divisio 6 Quotit Domitor umrtor Rmidr Lt 8 6 H A B Comprig cofficits of li powrs of i th umrtors o both sids : A + C 8 : A + B.5C 6 :.5A.5B which giv A 8, B, d C 6, d 8 H C DSP-4 Z 4 of 84 Dr. Rvi Bill

25 I ATLAB Th prtil frctios my b computd by usig th rsidu fuctio: 8 4 R R R H K p p p W dfi th cofficit vctors b [8, -4,, -] d [, -.5,.75, -.5]; R rprsts th rsidus prtil frctio costts, p th pols d K costt. ot tht i th umrtor ms, d i th domitor ms ; sic is ot lss th this is ot propr rtiol fuctio, so tht K will hv oro lmts. %Prtil frctios b [8, -4,, -], [, -.5,.75, -.5], [R, p, K] rsidu b, Th ATLAB rsults rturd r R -8 -i -8 +i 8 p.5 +.5i.5 -.5i.5 K 6 Thrfor, 8 H DSP-4 Z 5 of 84 Dr. Rvi Bill

26 DSP-4 Z 6 of 84 Dr. Rvi Bill Ivrs -trsform wh thr r rptd roots With rptd roots, tht is, -th ordr pol t w hv X i th form X, ROC > Th tbl blow givs th ivrs -trsforms for svrl vlus of d for th grl cs of rbitrry. Rptd Roots X x ʓ - [X] for ROC > u! u! u 4! u!... u Grl procdur for prtil frctio xpsio Sic X/ must b rtiol, it ts th form X L L L L K K K K If K < L th o dustmt is dd. Th prtil frctio xpsio is strightforwrd. If K L th divid util th rmidr polyomil i hs dgr of L or lss: X c K L K L + +c + c d d d L L L L L L Th first prt of th bov xprssio, c K L K L + +c + c, will vtully cotribut δ fuctios to th output squc som of which r tim-dvcd so tht th rsultig x will b ocusl. Th scod prt th propr frctio is xpdd ito prtil frctios. Assum w hv o rptd pol of ordr m, cll it, d tht ll th rst r distict, cll thm m+, m+,, L. Th lt d d d L L L L L L m m A + m m A + + A + L m B Th cofficits A m of thm d B m L of thm r foud s follows: A! m m m m d d,,,, m

27 B [ ], m+, m+,, L I th rsultig x th cotributio of th A trms is umbr of xpotils multiplid by,,, tc., d th cotributio of th B trms is umbr of complx xpotils. Exmpl 4.5. Fid th ivrs of H.5, > Solutio This trsform is propr rtiol fuctio. W shll us this xmpl to giv summry of th thr styls of obtiig prtil frctios: Expdig H dirctly, Expdig H / d Expdig H s i ATLAB lso itr. Wh th pols of H r distict th prtil frctio cofficits rturd by th ATLAB fuctio rsidu r th sm s i xpdig H /. Howvr, wh thr r rptd pols it ms diffrc i th cofficits s wll s i th fil lyticl forms of th ivrs trsforms i ths two mthods. I dditio, dirctly xpdig H rsults i lyticl form tht is still diffrt from th othr two. I y vt th thr ivrs trsforms r th sm s fr s th ctul squc vlus r cocrd. Expdig H W hv A B B H Thr is pol t d rptd pol t.5. Th cofficits A, B d B r giv by A B. 5.5 B d d.5 Thus H 4.5 Tig th ivrs -trsform, + h 4 ʓ - 4 ʓ - Expdig H / W hv ʓ ʓ -. 5 ʓ - 4 u.5! + 4 ʓ u.5 u u 4.5 u 8.5 u DSP-4 Z 7 of 84 Dr. Rvi Bill

28 H.5.5 Th cofficits C, D d D r giv by Thus C.5.5 D. 5.5 D H H d d Tig th ivrs -trsform, + D C ʓ - h 4 ʓ - 4 ʓ u +.5 u 4.5 u! 4 u 4.5 u 4.5 u.5 4 D +.5 Expdig H s i ATLAB W strt with H xprssd s rtio of polyomils i gtiv powrs of. Howvr, for th s of cotiuity w hv H H.5.5 E F W hv ordrd th cofficits i th ordr i which ATLAB displys thm. W c dfi v so tht corrspods to v d.5 to v. Th trsform ow pprs s Hv + + v.5 v v.5v Th cofficits E, F d F r giv by E.5 v F E.5. 4 F 4 F.5v F DSP-4 Z 8 of 84 Dr. Rvi Bill

29 Thrfor, F d dv 4 H v v 4 Tig th ivrs -trsform, v v.v v. v v h 4 ʓ ʓ u 4.5! 4 u 4.5 u u! 4 u u I ATLAB This prticulr st of prtil frctios my b computd by usig th rsidu fuctio: E F F H K p p p W dfi th cofficit vctors b [,, ] d [,,.5,.5]; R [E, F, F ] rprsts th rsidus yd to th bov prtil frctio cofficits, p th pols d K costt. %Prtil frctios b [,, ], [,,.5,.5], [R, p, K] rsidu b, Th ATLAB rsults rturd r R i -4 - i p i i K [] Thrfor, H DSP-4 Z 9 of 84 Dr. Rvi Bill

30 which grs with th hd-clcultd rsults. Exmpl 4.5. Fid th ivrs of X 6 6, for > ½ 5 / 4 / /6 4 X /6 A B + B C + + / / 4 / / / 4 /6 A / / 4 C B B X d d /6 / / 4 /6 / 4 / /6 /6 / / 4 / 4 / 4 / DSP-4 Z of 84 Dr. Rvi Bill / 4 / 4 / 4 /6 9 / 4 / / / / /6 5/ / / / 4 d d /6 / 4 / / 4 /6 / 4 / 4 / 9 5 / + 5 X + Tig th ivrs -trsform, x ʓ - {X} / / 9 + / + 9 / 9 / / 4 ʓ - {} + 5/ʓ - 9ʓ - / / + 9ʓ - / 4 5 δ + u 9 u + 9 u 4 Othr possibilitis If w choos to xpd X, rthr th X /, ito prtil frctios, w d to prform log divisio to rduc th dgr of th umrtor by rsultig i / 4 / X + + X 5/ 4 / /6 whr X is th propr frctio prt of th bov X / 4 / 5/ 4 / /6

31 Eithr X itslf or X / my ow b xpdd ito prtil frctios. I ATLAB Th prtil frctios my b computd by usig th rsidu fuctio: /6 R R R X K + + 5/ 4 / /6 p p p W dfi th cofficit vctors b [, -,, -/6] d [, -5/4, /, -/6]; R rprsts th rsidus prtil frctio cofficits, p th pols d K costt. ot tht i th umrtor ms, d i th domitor ms ; sic is ot lss th this is ot propr rtiol fuctio, so tht K will hv oro lmts. %Prtil frctios b [, -,, -/6], [, -5/4, /, -/6], [R, p, K] rsidu b, + Th ATLAB rsults rturd r R p K Thrfor, /6 X 5/ 4 / / Exmpl 4.5. Fid th ivrs of X / / 4 X DSP-4 Z of 84 Dr. Rvi Bill A A for ROC > ½. + + / / 4 / / / / A / 4 / / / 4 A A d! d / 4 / d! d / 4 / B / / 4 X 6 / 8 / / / 4 A B + / 4

32 X / / / x ʓ - {X} 6ʓ - ʓ / - / + 8ʓ - 8 ʓ / - / 4 6 u u + 8 u! Th u my b fctord out tc. / 4 8 u 4 ATLAB X 4 7/ 4 9/8 5/6 / %Prtil frctios b [,,, ], [, -7/4, 9/8, -5/6, /], [R, p, K] rsidu b, Th ATLAB rsults rturd r R.+ *.44 +.i i p.5 +.i.5 -.i.5.5 K [] X 44 / 88 + / 4 + / 8 / 4 + DSP-4 Z of 84 Dr. Rvi Bill

33 Rltioships mog systm rprsttios A discrt-tim lir shift-ivrit systm c b chrctrid by its uit smpl rspos, diffrc qutio, systm fuctio, or frqucy rspos. Assum tht systm is dscribd by th lir costt cofficit diffrc qutio y br x r r Systm fuctio T th -trsform of both sids of th bov qutio ʓ y ʓ br x r, or r ʓ Y b r r y Y br Th systm fuctio is H r r Xb r r Y X ʓ x r, or X, or r r b r r Uit smpl rspos If x δ th X ʓ[x] ʓ[δ]. Th corrspodig y is th uit smpl rspos h. W hv Y H, or Y H.X H. H X So, giv H, th systm fuctio, th uit smpl rspos is h ʓ - [H]. Th diffrc qutio from th H Th systm fuctio H is first writt i trms of Y gtiv powrs of d st qul to. Th cross-multiply d t th ivrs -trsform X to gt th diffrc qutio. Frqucy rspos of th systm is th Fourir trsform DTFT of th uit smpl rspos h: H h Compr this with th systm fuctio H dfid s th -trsform of th uit smpl rspos h H h Thus th frqucy rspos, if it xists, c b obtid by rplcig th i H by follows: s DSP-4 Z of 84 Dr. Rvi Bill

34 H h H Th systm is implicitly BIBO-stbl. Th bov rltioships for stbl, cusl systm rprstd by lir costt cofficit diffrc qutio r summrid i digrm blow. St H Diffrc Equtio T -trsform, solv for Y/X H T ivrs -trsform h Writ i trms of, cross multiply, t ivrs -trsform T -trsform Exmpl 4.6. Fid th impuls rspos of y y + x. Solutio ot tht w hv solvd this i th tim domi rlir. Tig th -trsform of both sids with ro iitil coditios, Y Y + X, or Y H X Assum cuslity. Th from th tbl of trsforms, h ʓ - [H] u. Cuslity i trms of th -trsform, H, d th ROC A cusl LTI systm hs impuls rspos h for <, d is, thrfor, right-sidd squc. This lso implis tht th ROC of H is th xtrior of circl i th -pl. For cusl systm th powr sris H h h + h + h + Eq. dos ot iclud y positiv powrs of. Cosqutly, th ROC icluds. Thrfor, w hv th pricipl: A discrt-tim LTI systm is cusl if d oly if th ROC of its systm fuctio is th xtrior of circl, d icluds. Th iitil vlu thorm sys tht for cusl squc, h, th iitil vlu is giv by h lim H This my b s by sttig i Eq. mig ll trms go to ro xcpt th trm h. Thus, for cusl squc, h, if h is fiit, th, lim H is fiit. Cosqutly, with H xprssd s rtio of polyomils i positiv powrs of, th ordr of th umrtor polyomil cot b grtr th th ordr of th domitor polyomil if it wr thr DSP-4 Z 4 of 84 Dr. Rvi Bill

35 DSP-4 Z 5 of 84 Dr. Rvi Bill would b positiv powrs of i th powr sris of H, corrspodig to o-ro h for gtiv ; lso would ot b icludd i th ROC; or, quivltly, th umbr of fiit ros of H cot b grtr th th umbr of fiit pols. Th bov discussio is summd up s follows: A discrt-tim LTI systm with rtiol systm fuctio H is cusl if d oly if. Th ROC is th xtrior of circl outsid th outrmost pol, d,. With H xprssd s rtio of polyomils i, positiv powrs of, th ordr of th umrtor is ot grtr th th ordr of th domitor. Coditio lo is ot ough bcus th squc my b right-sidd but ot cusl. If H is rprstd s rtio of polyomils i s H L L L L K K K K Eq. th L K if th systm is cusl i othr words domitor dgr umrtor dgr. O th othr hd, if w writ H s th rtio of polyomils i gtiv powrs of s H b b b b / /... / / /... / b b b b th, if th systm is to b cusl,. This is s by sttig, d rquirig tht h b / b fiit. This is illustrtd with xmpl whr,.g., H which, by log divisio, c b s to coti positiv powr of hc o-csul. ot Wh H is writt s rtio of polyomils i positiv powr of, s i Eq., w hv rquird tht L K for cuslity. Ths L d K r ot to b cofusd with th d cotid i th diffrc qutio. Cosidr, for xmpl, th systm y + y x + b x whr, ccordig to th ottio of th diffrc qutio, d. Apprtly is grtr th d this is llowbl. I othr words, thr is o rstrictio o th rltiv vlus of d. For, th trsfr fuctio is giv by H X Y b b b d it is s tht th umrtor dgr K is ot grtr th th domitor dgr L. Thus th systm is cusl. As othr xmpl cosidr y + y x + b x+ which is o-cusl bcus of th x+ trm. Th trsfr fuctio is H X Y b b b b ot tht, wh th umrtor d domitor r xprssd i trms of gtiv powrs of,. O th othr hd, wh th umrtor d domitor r xprssd i trms of positiv powrs of, w hv

36 DSP-4 Z 6 of 84 Dr. Rvi Bill H X Y b with th umrtor dgr grtr th th domitor dgr. Omit Rtiol trsfr fuctio; LTI systm Giv th systm with th th ordr diffrc qutio, y + y + + y b x + b x + + b x, w my writ it i th mor compct form y r r r x b, ot tht som uthors t th cofficit of y,, to b. I th bov diffrc qutio w my divid through by so tht th cofficit of y is. W c fid th trsfr fuctio of th systm by tig th -trsform o both sids of th qutio. W ot tht i fidig th impuls rspos of systm d, cosqutly, i fidig th trsfr fuctio, th systm must b iitilly rlxd ro iitil coditios. Thus, if w ssum ro iitil coditios, w c us th lirity d tim-shift proprtis to gt Y X r r b r so tht H X Y r r r b Eq. Th corrspodig impuls rspos c b foud s h ʓ {H}. Th pols of th systm trsfr fuctio r th sm s th chrctristic vlus of th corrspodig diffrc qutio. For th systm to b stbl, th pols must li withi th uit circl i th -pl. Cosqutly, for stbl, cusl fuctio, th ROC icluds th uit circl. Th systm fuctio, H, is rtiol fuctio: H D b b b b b Hr d D std for umrtor d domitor rspctivly. If d b, w c void th gtiv powrs of by fctorig out b d s follows: H D b. /... / /... / b b b b Sic d D r polyomils i, thy c b xprssd i fctord form s H D b p p p

37 C., whr C b / p Thus H hs fiit ros t,,,, d fiit pols t p, p,, p, d ros if > or pols if < t th origi. Pols d ros my lso occur t. A pol xists t if H, d ro xists t if H. If w cout th pols d ros t d s wll s th pols d ros, w fid tht H hs xctly th sm umbr of pols d ros. By dfiitio th ROC of H should ot c ot coti y pols. Propr rtiol fuctio Tig, w hv b b b... b H D... This is clld propr rtiol fuctio if d <. This mouts to syig tht th umbr of fiit ros is lss th th umbr of fiit pols. Fiit ros d pols xclud thos t. This coditio is rltd to prtil frctio xpsio d hs othig to do with cuslity. Ed of Omit Exmpl 4.6. Giv th pol-ro plot for H Solutio Th domitor hs roots pols t, d.6 Thr is ro t. Furthr, sic th domitor dgr is grtr th th umrtor dgr by it is clr tht H, so tht thr is dditiol ro t. I ATLAB th trsfr fuctio is spcifid s rtio of polyomils i. H.. Th umrtor cofficits, {b i, i to } d th domitor cofficits { i, i to } r spcifid s th two vctors b [, ] d [, -, -]. %Pol-ro plot b [, ]; [, -, -]; pl b, DSP-4 Z 7 of 84 Dr. Rvi Bill

38 Imgiry Prt Rl Prt DSP-4 Z 8 of 84 Dr. Rvi Bill

39 DSP-4 Z 9 of 84 Dr. Rvi Bill Exmpl 4.6. Giv th pol-ro plot for H Solutio From H w c s tht thr r 9 pols t d 8 ros t sudry plcs d dditiol ro t owig to th domitor dgr big grtr th th umrtor dgr by. For th ATLAB sgmt th umrtor d domitor cofficits r t from H %Pol-ro plot b [: 9]; [, ]; pl b, Rl Prt Imgiry Prt

40 Imgiry Prt Exmpl Giv th pol-ro plot for y x +.8 x.8 x.45 y Solutio Th ros r,.8 Th pols r giv by, ± d.99.45]. For th ATLAB progrm th cofficit vctors r b [,.8, -.8] d [,, %Pol-ro plot b [,.8, -.8]; [,,.45]; pl b, Rl Prt DSP-4 Z 4 of 84 Dr. Rvi Bill

41 From th grl form H i Eq. w c obti two importt spcil forms: th ll-ro systm, d th ll-pol systm. Thr r, of cours, trivil pols or ros prst. Th ll-ro systm If for, w hv H b r r r. Eithr t or cosidr tht is bsorbd i th b r cofficits, so tht b b... b H b b b... b I this cs, H cotis ros d th ordr pol t th origi. Sic th systm cotis oly trivil pols t d o-trivil ros, it is clld ll-ro systm. Such systm hs fiit-durtio impuls rspos FIR, d is clld FIR systm or movig vrg A systm. ot tht th corrspodig dfrc qutio is y b x + b x + + b x Th ll-pol systm O th othr hd, if b for, w hv b H b... b. / /... / Hr gi, ithr t or imgi tht it is bsorbd i th othr cofficits vi., b,,,,. Thus b H... Hr H hs pols d th ordr ro t th origi. W usully do ot m rfrc to ths trivil ros. As rsult this systm fuctio cotis oly o-trivil pols d th corrspodig systm is clld ll-pol systm. Du to th prsc of th pols, th impuls rspos of such systm is ifiit i durtio, d hc it is IIR systm. W c divid th umrtor ito th domitor d thrby xpd H ito ifiit sris from which it is vidt tht h is of ifiit durtio. ot tht th corrspodig dfrc qutio is y + y + + y b x Th pol-ro systm Th grl form, though, cotis both pols d ros d th systm is clld pol-ro systm with pols d ros, b b b... b H... Pols d/or ros t d r implid but r ot coutd xplicitly. Du to th prsc of pols, th pol-ro systm is IIR systm. 4.7 Ivrs -trsform by powr sris xpsio log divisio If th -trsform is xprssd s rtiol fuctio rtio of polyomils i or w c us log divisio to xpd it ito powr sris. If th trsform is xprssd s irrtiol fuctio w c us th pproprit powr sris xpsio formul vilbl i mthmticl DSP-4 Z 4 of 84 Dr. Rvi Bill

42 tbls such s th CRC Tbls. ot tht if th trsform is xprssd s irrtiol fuctio th th prtil frctio xpsio mthod of ivrsio wo t wor. By dfiitio th -trsform of th squc x is giv by X x + x + x + x + x + This is powr sris Lurt sris. So by log divisio w obti th powr sris xpsio of X d th, by compriso with th powr sris dfiitio giv bov, w c idtity th squc x. I prticulr th cofficit of is th squc vlu x. Th mthod is usful i obtiig quic loo t th first fw vlus of th squc x. This pproch dos ot ssur lyticl solutio. Th ROC will dtrmi whthr th sris hs positiv or gtiv xpots. For right-sidd squcs th X will b obtid with primrily gtiv xpots, whil lft-sidd squcs will hv primrily positiv xpots. For ulr ROC, Lurt xpsio would giv both positiv- d gtivtim trms. This lst possibility is illustrtd i th xmpl blow by tig littl hlp from prtil frctios. Exmpl 4.7. Fid th ivrs trsform, by log divisio, of X whr th ROC is >, b <, c < < Solutio ROC is >. W xpct right-sidd squc, with prdomitly gtiv xpots of. For th log divisio rrg umrtor d domitor s dcrsig powrs Im ROC R of d th divid; or s icrsigly gtiv powr of i.., d th divid Q D DSP-4 Z 4 of 84 Dr. Rvi Bill

43 Thus X By compriso with th dfiig qutio X x + x + x + x + w s tht th squc vlus r x x, or x for <, d x, x, x 5, tc. Altrtivly, it is lso possibl to writ X s rtio of polyomils i X ot tht th polyomils r writt i th ordr of icrsig gtiv powrs of, tht is,. Log divisio givs th sm swr s obtid rlir: Q D Solutio b Th ROC is <. W xpct lft-sidd squc with prdomitly positiv xpots of. For th log divisio th polyomils r writt i th ordr of icrsig powrs of or dcrsigly gtiv powrs of, i..,. ROC Im R / 5/4 9/8 Q D / / 5/ + / 5/ + 5/4 5/4 4 9/4 + 5/4 4 9/4 + 7/8 4 9/8 5 Thus X / 5/4 9/8 9/8 5/4 /. By comprig with th dfiig qutio X x + x + x + x + x + DSP-4 Z 4 of 84 Dr. Rvi Bill

44 w s tht th squc is giv by x /, x 5/4, x 9/8, tc., d x for Th othr wy of log divisio is show blow: / 5/4 9/8 Q D / / Omit Solutio c Th ROC is < <. W xpct two-sidd squc with both positiv d gtiv xpots of. Looig t th pol-ro cofigurtio, th pol t implis right-sidd squc d th pol t lft-sidd squc. Obviously ust sigl log divisio cot giv both th lft-sidd d th right-sidd squcs simultously. W shll obti th prtil frctio xpsio first d th procd with th divisio to obti th squcs sprtly. Ths two squcs r th combid ito o squc to gt th solutio. ot tht w do this oly to illustrt th mthod of log divisio. But oc w us prtil frctios th utility of log divisio is ullifid. Im ROC R X A B X X A B +. /. / + + For th trm w hv right-sidd squc giv by log divisio thus: DSP-4 Z 44 of 84 Dr. Rvi Bill

45 Q D Th corrspodig squc is x R,, othrwis For th trm w hv lft sidd squc / /4 /8. Q D + / / / /4 /4 /4 /8 4 /8 4 Th corrspodig squc is x L, <, othrwis Th complt squc is th x x R + x L,, < Ed of Omit 4.8 Computtio of frqucy rspos Lt th systm fuctio b giv by H r b r r Th frqucy rspos is H or Hω H. Thus DSP-4 Z 45 of 84 Dr. Rvi Bill

46 whr H Hω H b cos r r r r r cos b r r r b si r si b cos r r r r cos A B C D b si r r si A b r cos r, B b r si r, C cos, D si. r r Th mgitud d phs of H r giv, rspctivly, by A B Hω C D d H B D t t A C Thorm Th frqucy rspos H for BIBO-stbl systm will lwys covrg. Accordigly vry BIBO-stbl systm will hv frqucy rspos d dscribbl stdystt rspos to siusoidl iputs. But, th covrs of this sttmt is ot tru, tht is, th fct tht H xists dos ot imply tht th systm is stbl. Exmpl 4.8. [Th idl low pss filtr] For th Hω giv i figur blow fid h, th uit smpl rspos. Hω Priodic π ω c ω c π π ω <Hω Phs Priodic Solutio Th uit smpl rspos is th ivrs DTFT of Hω h π ω c ω c π π si H c c d d, for ll c c c ω c c DSP-4 Z 46 of 84 Dr. Rvi Bill

47 It is s tht h for gtiv so tht th idl low pss filtr is ocusl. orovr, lthough h tils off s gos from to d from to, it c b show tht h is ot fiit. This ms tht th idl low pss filtr is ot BIBO-stbl ithr. Exmpl 4.8. [] A discrt systm is giv by th diffrc qutio y 5 y x + 4 x whr x is th iput d y is th output. Dtrmi th mgitud d phs rspos s fuctio of frqucy for ω π. ot tht th systm is ot stbl sic it hs pol t 5, which is outsid th uit circl. Th fct tht th stdy stt frqucy rspos xists dos ot m tht th systm is stbl. Solutio [S lso Uit I] Tig th -trsform d with dos of lgbr w fid th trsfr fuctio Y 4 H X 5 Th frqucy rspos is giv by 4 4 Hω H 5 5 D ω 4 cos 4 si ω ω cos 4 si d si t cos 4 Dω D 5 cos 5 si Dω cos 5 si d D si t cos 5 Dω Hω D cos 4 si cos 5 si H D si t si cos 4 t cos 5 Th frqucy rspos c b plottd. ot tht Hω is v fuctio d H is odd fuctio of ω. Usig ATLAB: 4 4 b b b... Hω Hr th vctors b d spcify, rspctivly, th umrtor d domitor cofficits. I our xmpl b, b 4,, d 5. Th ATLAB sgmt d th corrspodig plots follow. ot tht th plot gos from to. Compr with th solutio obtid i Uit I usig diffrt fuctio. b [, 4]; %umrtor cofficits [, -5]; %Domitor cofficits w -*pi: pi/56: *pi; [h] frqb,, w; subplot,,, plotw, bsh; xlbl'frqucy \omg', ylbl'gitud'; grid DSP-4 Z 47 of 84 Dr. Rvi Bill

48 gitud Phs - Rdis subplot,,, plotw, glh; xlbl'frqucy \omg', ylbl'phs - Rdis'; grid Frqucy Frqucy Exmpl 4.8. Assum H is cusl systm. Fid th diffrc qutio d 6 th frqucy rspos. Solutio Arrg H i trms of gtiv powrs of Y H X 6 6 Cross multiplyig Y 6 X 6Y Y Y X X Tig th ivrs -trsform 6y y y x x y 6 y + 6 y + x 6 x Th pols of H r loctd t ,.5 d / 6 d r isid th uit circl. This big cusl systm, th ROC is > ½ d cotis th uit circl. Th systm is stbl, d th frqucy rspos is migful. It is giv by DSP-4 Z 48 of 84 Dr. Rvi Bill

49 whr Hω H 6 6 ω cos si cos si si t cos Dω 6 6 6cos 6si cos si 6cos cos 6si si Th mgitud rspos is giv by cos si Hω 6cos cos 6si si Th phs rspos is giv by si 6si si H t t cos 6cos cos D 6si si t 6coscos Usig ATLAB: H 6 6 b b b... Hω 6... Hr th vctors b d spcify, rspctivly, th umrtor d domitor cofficits. I our xmpl b, b, b, 6, d. Th ATLAB sgmt d th corrspodig plots follow. ot tht th plot gos from to. b [,, -]; %umrtor cofficits [6, -, -]; %Domitor cofficits w -pi: pi/56: pi; [h] frqb,, w; subplot,,, plotw, bsh; xlbl'frqucy \omg Rd', ylbl'gitud'; grid subplot,,, plotw, glh; xlbl'frqucy \omg Rd', ylbl'phs - Rdis'; grid DSP-4 Z 49 of 84 Dr. Rvi Bill

50 Phs - Rdis gitud Frqucy Rd Frqucy Rd Exmpl [] Discuss th stbility of H ssumig it is cusl systm. Fid th diffrc qutio d th frqucy rspos. b Dtrmi th frqucy, mgitud d phs rsposs d tim dly for th systm y + /4 y x x. Solutio Fid th ROC d th pols: H Thr is ro t. Th domitor hs roots t, d.6 Uit circl Im ROC >.6 R.6.6 DSP-4 Z 5 of 84 Dr. Rvi Bill

51 Th pol loctios r show hr. For th systm to b cusl th ROC is th xtrior of circl with rdius.6. I this cs ROC dos ot iclud th uit circl. Equivltly, ll th pols do ot li withi th uit circl. Hc th systm is ot stbl. b Tig th -trsform o both sids of y + /4 y x x w gt Y + /4 Y X X, or Y { + /4 } X { }, or Y H X / 4.5 Thr is sigl ro t d sigl pol t.5 which is isid th uit circl hc stbl. Th frqucy rspos is giv by Hω H.5.5 D whr ω cos si cos si Dω. 5 cos si. 5 cos.5 si Th mgitud rspos is giv by Hω cos si cos.5 si si t cos.5 si t cos Th phs rspos is giv by si si H t t cos cos.5 d Th tim group dly is giv by d H. Usig ATLAB: b b b... Hω Hr th vctors b d spcify, rspctivly, th umrtor d domitor cofficits. I our xmpl b, b,,.5. Th ATLAB sgmt d th corrspodig plots follow. ot tht th plot gos from to. b [, -]; %umrtor cofficits [,.5]; %Domitor cofficits w -pi: pi/56: pi; [h] frqb,, w; subplot,,, plotw, bsh; xlbl'frqucy \omg Rd', ylbl'gitud'; grid subplot,,, plotw, glh; xlbl'frqucy \omg Rd', ylbl'phs - Rdis'; grid DSP-4 Z 5 of 84 Dr. Rvi Bill

52 Phs - Rdis gitud Frqucy Rd Frqucy Rd 4.9 Z-trsforms with iitil coditios To solv th th ordr diffrc qutio y y + br x r r with o-ro iitil coditios w d iitil coditios o th output y d iitil coditios o th iput x. Usully th iput is pplid suddly i.., it is stppd ito th systm t, so tht o iitil coditios r dd for it, tht is, x for <. Th output y, howvr, i grl, will hv o-ro iitil coditios for to. W r solvig for y for, so tht Y ʓ{y} is th o-sidd -trsform. Th diffrc qutio cotis othr trms li y, y, tc. which r dlyd vrsios of y. Suppos, th w shll hv y, y, d y to dl with. Th trsform of y is hdld s follows. First, for th squc y s show blow w dfi Y + ʓ{y, } A B C... W shll rfr to this loosly s ust Y wh thr is o possibility of cofusio. y y y y A B C DSP-4 Z 5 of 84 Dr. Rvi Bill

53 W th obti y by dlyig th squc by o uit, show blow. y y A B C As c b s from th grph ʓ{y, } y A B C... Y y Y + I similr fshio 4 ʓ{y, } y y A B C... Y y y d by xtsio ʓ{y, } Y y y y For this would b th lst. But w c grli ʓ{y, } Y y y... y I th cs of th iput x, sic it is pplid suddly t, th iitil coditios r ro, tht is, x x x, so tht ʓ{x u} X With this ituitiv bcgroud w giv blow th mthmticl drivtio of th - trsform of th dlyd tructd squc. Z-trsform of dlyd tructd squc iitil coditios Th o-sidd -trsform of x is X + ʓ{x u} x Giv th squc x, w dly it by uits, d th truct it to th lft of to gt x u. W wt fid th -trsform of x u. ʓ{x u} x u x If w lt r, th r+, d th summtio limits to bcom r to. Th ʓ{x u} x r r r r x r r r x r x r r X x r r r r DSP-4 Z 5 of 84 Dr. Rvi Bill X + IC r Y +

54 DSP-4 Z 54 of 84 Dr. Rvi Bill... x x x X... x x x X W shll loosly rfr to X + s X d writ th rsult s ʓ{x u}... x x x X Th bov rsult is usd to solv lir costt cofficit diffrc qutios with iputs tht r stppd ito systm. Suppos w wt th solutio of y r r r x b, subct to th iitil coditios {yi, i,,, } d {xi, i,,, } W t th -trsform of th qutio usig th rsult drivd bov for dlyd-tructd squcs ʓ y ʓ r r r x b, } { y r r r x b } {, whr w hv usd Z to m ʓ th -trsformtio oprtio. Th lft hd sid is LHS ʓ{y} + ʓ{y } + ʓ{y } + + ʓ{y } Y + { Y + y } + { Y + y + y } + + { Y + y + y + + y + } ot tht i trms of th drivtio rlir ll of th Y s r Y + s, i.., o-sidd trsforms. All th Y trms c b groupd togthr udr summtio, d ll th rmiig trms, du to th iitil coditios {yi, i,,, }, c b groupd togthr so tht th bov c b writt s LHS Y + g{, y, y,, y } By followig similr procdur th right hd sid c lso b writt s follows hr gi th X s r X + s, i.., o-sidd trsforms: RHS r r r X b + h{, x, x,, x } Du to Iitil Coditios Du to Iitil Coditios Iitil coditio trms Iitil coditio trms

55 Writig out i full, LHS RHS bcoms Y + g{ } br r Fctorig out Y d X d rrrgig w hv Y Y X X b r r r b r r + r r X + h{ } + h{ } g{ } h{...} g{...} Tig th ivrs -trsform w gt r br y ʓ r X + ʓ h{...} g{...} To summri: to solv for y w t th -trsform of th lir costt cofficit diffrc qutio usig iitil coditios, mipult i th -domi to gt Y d th t th ivrs -trsform of Y to gt y. Exmpl 4.9. Fid th solutio to y y + y 4, with iitil coditios y 4, y. Solutio Thr r thr mthods of solutio:. Fid th itrtiv solutio i th discrt-tim domi. I grl this will ot giv lyticl closd form of solutio.. Solv i th discrt-tim domi homogous solutio + prticulr solutio.. Solv i th frqucy domi s w do blow. For iput squc x tht is stppd ito systm, spcifid i words li x for <, th iitil coditios r clrly ro d do ot mttr. But for output squc y whr th iitil coditios y, y r xplicitly giv to b o-ro w d to us th bov drivd -trsform for dlyd tructd squc. I prticulr w hv ʓ{y} Y ʓ{y } Y y ʓ{y } Y y y Tig th -trsform of th diffrc qutio w gt ʓ y y y ʓ 4 ʓ{y} ʓ{y } + ʓ{y } / 4 Y Y y + Y y y DSP-4 Z 55 of 84 Dr. Rvi Bill / 4

56 Pluggig i th iitil coditios y 4 d y Y Y 4 Y 4 + Y / 4 Y / 4 Y.5 Y / 9/ 4 / Y / 4 / 9 / 4 / / 4 9 / 4 / / 4 Y 9/ 4 / / 4 / 9/ 4 / A / / / 4 9/ 4 / B / / 4 / 9/ 4 / C / 4 / Y / / + + / 4 / Y + + / 4 / / 4 9 / 4 / / 4 DSP-4 Z 56 of 84 Dr. Rvi Bill A B C + + / 4 / y u 4 Th itrtiv solutio for this problm ws obtid i Uit I. Th tim-domi solutio ws covrd i HW Extr. Th solutio is rptd blow y,,,, Exmpl 4.9. [] Solv th followig lir diffrc qutio y + y 4 y giv y y. Solutio ot tht th output is turl rspos corrspodig to th spcifid iitil coditios. Thr is o forcd rspos sic x. Th itrtiv solutio is 7 y,,,,

57 For th solutio i th frqucy domi w t th -trsform of th diffrc qutio ʓ y y y ʓ 4 ʓ{y} + ʓ{y } 4 ʓ{y } Y + Y y 4 Y y y Y + Y 4 Y Y Y.5.5 Y Th domitor o th right hd sid hs roots t, Y A B A.9 DSP-4 Z 57 of 84 Dr. Rvi Bill 5.9 d.89 4 B Y.9.89 y , Th first fw vlus of th squc r y.5,.76,.5,.9,... d should b comprd with th itrtiv solutio. I th cotxt of ATLAB, w my us filtrb,, x to grt th squc y. Th cofficits of y. d x. r umbrd slightly diffrtly s blow: From th diffrc qutio y + y + y + b x + b x + b x + y + y 4 y, w ot tht th iput is x d th cofficits of y. d x. giv us th d b vctors: [,.5, -.5] d b []. Th o-ro iitil coditios y y must first b covrtd to quivlt iitil coditios for th filtr fuctio to wor. W spcify th vctor yic [y, y ] [, ] d grt th quivlt iitil coditios ic by th fuctio filticb,, yic. Th quivlt iitil coditios r th usd to grt th filtr output through filtrb,, x, ic. Th ATLAB sgmt follows:

58 x y %o-ro iitil coditios b [], [,.5, -.5], yic [, ], :5, x.* % %Equivlt iitil coditios ic filticb,, yic, y filtrb,, x, ic subplot,,, stm, x; xlbl'', ylbl'x'; titl'iput Squc'; subplot,,, stm, y; xlbl'', ylbl'y'; titl'output Squc'; Th output is: y [ ] Iput Squc Output Squc Exmpl 4.9. [] Fid th rspos squc for th filtr dfid by y 7 y + 6 y x Assum th systm is iitilly rlxd. Obti th systm fuctio d plot its pols d ros. Solutio Th phrs iitilly rlxd ms tht th iitil coditios r ro, tht is, y for < d x for <. Th qustio dos t spcify wht th iput x is, so ssum δ. Wht will th output b if both th iput d th iitil coditios r ro? 4. Stdy-stt d trsit rsposs for first ordr systm DSP-4 Z 58 of 84 Dr. Rvi Bill

59 W cosidr siusoidl iputs. Although th prsttio is oly for first ordr systm, th rltioship stblishd for th stdy-stt rspos i trms of th trsfr fuctio of th systm is grl rsult for stbl systms d siusoidl iputs. Th systm is y y + x, with th iitil coditio y d th iput x cos u. W hv cosidrd th timdomi bhvior of this systm i Uit I. Assum < i ordr to hv stbl systm. Th systm fuctio is obtid with ro iitil coditios, Y Y + X, or Y H X Th solutio of th diffrc qutio is obtid by tig th -trsform d usig th giv iitil coditio Y [ Y + y ] + X Y y + X Y y + X y H + X H cos Sic X ʓ{x} ʓ{ cos u}, w hv cos Y y H + Y Y cos H Y + Y cos Hr Y is th ro-iput rspos du to th iitil coditios y Y y H d Y is th forcd rspos du to th iput x cos Y cos H cos cos Y is lrdy i covit form for tig th ivrs, but Y must b xpdd ito prtil frctios s blow. Y cos cos Fctorig of cos * A B B + + cos A cos cos cos DSP-4 Z 59 of 84 Dr. Rvi Bill