www.tuworld.com www.tuworld.com 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
www.tuworld.com www.tuworld.com 4. 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
www.tuworld.com www.tuworld.com 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
www.tuworld.com www.tuworld.com 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
www.tuworld.com www.tuworld.com 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
www.tuworld.com www.tuworld.com 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
www.tuworld.com www.tuworld.com 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
www.tuworld.com www.tuworld.com 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
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 www.tuworld.com www.tuworld.com
www.tuworld.com www.tuworld.com 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
www.tuworld.com www.tuworld.com 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
www.tuworld.com www.tuworld.com 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
www.tuworld.com www.tuworld.com 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
www.tuworld.com www.tuworld.com 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 6 + 4 4 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
www.tuworld.com www.tuworld.com 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
www.tuworld.com www.tuworld.com H Y X.5 /.5 /.5 Th domitor hs roots t 4...5,.5.5.5 +.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 >.5 +.5 or >.77. Figur Im Uit circl Pol t.5 +.5 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.5.5.5.5.5 A +.5.5 Solvig for A d A *, w gt A.5.5.5.5 * A.5.5.5.5.5. 5.5.5.5.5 t.5.5 5 / A *.5 +.5 Thus w hv * H A A +.5.5.5.5 H A + A *.5.5.5.5 b DSP-4 Z 6 of 84 Dr. Rvi Bill
www.tuworld.com www.tuworld.com whr.5 +.5.5.5 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 / 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
www.tuworld.com www.tuworld.com 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 +.5.5.5.5 W hv, i ffct, rrgd H s H + Thrfor, h ʓ - {H} ʓ - + ʓ.5 -. 5 DSP-4 Z 8 of 84 Dr. Rvi Bill
www.tuworld.com Imgiry Prt www.tuworld.com 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,.8.6.4. -. -.4 -.6 -.8 - - -.5.5 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 + +.5 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
www.tuworld.com www.tuworld.com [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 +.5.5.5.5.5.5.5 + 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 - /
www.tuworld.com www.tuworld.com 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 4.5.4 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 + + 4 / / A B C / / / X + / X + / x δ + u / u Exmpl 4.5.5 [] Fid th ivrs -trsform of X...56 Solutio Th roots of th qudrtic i th domitor r giv by u
www.tuworld.com www.tuworld.com, b b 4c...4 X.8.7 X A B C....5 A B C + +.8.7. 8. 7.8.7.7.8.8.7 / /.56.79.75 4.56..5.8 d.7 X.79 /.75 + +.8.7 /.75 X.79 + +.8. 7 x ʓ - {X}.79 δ +.8 u +.75.7 u Exmpl 4.5.6 Fid th ivrs -trsform of X / / 4 Solutio X / / 4 A / / 4 B C / 4 / / / 4 8 6 A B C + + / / 4 8 X 8 8 6 + / / 4 8 6 X 8 + / / 4 x 8 δ + 8 u 6 u 4, > ½. DSP-4 Z of 84 Dr. Rvi Bill
www.tuworld.com www.tuworld.com 4 Exmpl 4.5.7 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 + / / 4 + 5 7 x δ+ + + u 4 6 4 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 4.5.8 Assumig tht H idpdtly of ch othr 4 5 is cusl systm fuctio, prov th followig
www.tuworld.com www.tuworld.com Solutio 9 +. 5 h 4/5 + 9/5 b h 5 u + 4 h + 9 H 4 / 5 9/5 + 5 5 5 5 u u- u-, b H + 5 4 5, c By log divisio H Exmpl 4.5.9 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 6 8 + / 4 + 6 / Howvr, w my lso procd with gtiv powrs of s blow w my viw p s w vribl: 8 4 8 4 / 4 /.5.5 H 8 4.5.75.5 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.5 +.75.5 + + 4 + 8 umrtor + + 6 + 6 8 Rmidr Lt 8 6 H 6 +.5.75.5 8 6.5.75.5 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 6 +.5 C +.5.5 6 +.5 DSP-4 Z 4 of 84 Dr. Rvi Bill
www.tuworld.com www.tuworld.com I ATLAB Th prtil frctios my b computd by usig th rsidu fuctio: 8 4 R R R H K + + +.5.75.5 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 6 +.5.5 8.5.5 + 8.5 + DSP-4 Z 5 of 84 Dr. Rvi Bill
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 www.tuworld.com www.tuworld.com
www.tuworld.com www.tuworld.com 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, >.5.5.5 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.5 + +.5.5.5.5 Thr is pol t d rptd pol t.5. Th cofficits A, B d B r giv by A.5.5. 5 B. 5.5 B d d.5 Thus H 4.5 Tig th ivrs -trsform, + h 4 ʓ - 4 ʓ - Expdig H / W hv 4.. 5 4 +.5 ʓ -.5.5.5 4.5 + 4 ʓ -. 5 ʓ - 4 u.5! + 4 ʓ -.5.5 u.5 u + 4 4 u 4.5 u 8.5 u DSP-4 Z 7 of 84 Dr. Rvi Bill
www.tuworld.com www.tuworld.com H.5.5 Th cofficits C, D d D r giv by Thus C.5.5 D. 5.5 D H H d d 4.5.5.5.5 4. + +.5 4 Tig th ivrs -trsform, + D C +. 5.5 4 4.5 + ʓ - h 4 ʓ - 4 ʓ -.5. 5 4 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.5.5.5.5 H.5.5 E F +.5 +.5 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
www.tuworld.com www.tuworld.com Thrfor, F d dv 4 H v v 4 Tig th ivrs -trsform, v v.v v. v v 4 +.5 + h 4 ʓ - 4.5.5 4 4 ʓ -.5 4 u 4.5! 4 u 4.5 u 4.5.5 u! 4 u + 8.5 u I ATLAB This prticulr st of prtil frctios my b computd by usig th rsidu fuctio: E F F H K + + +.5.5 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 4 - + i -4 - i p - -.5 + i -.5 - i K [] Thrfor, H.5.5 4 +.5 + 4 +.5 DSP-4 Z 9 of 84 Dr. Rvi Bill
www.tuworld.com www.tuworld.com 4 4 +.5 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 + 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
www.tuworld.com www.tuworld.com 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 -4. 5. 9. p.5.5.5 K Thrfor, /6 X 5/ 4 / /6 9 +.5 4.5 + 5 +.5 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 / 6 8 + + / 8 8 + / 4 A B + / 4
www.tuworld.com www.tuworld.com X 6 + 8 8 / / / 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 -.88 -.i.4 -.8 p.5 +.i.5 -.i.5.5 K [] X 44 / 88 + / 4 + / 8 / 4 + DSP-4 Z of 84 Dr. Rvi Bill
www.tuworld.com www.tuworld.com 4.6 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
www.tuworld.com www.tuworld.com 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
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 www.tuworld.com www.tuworld.com
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 www.tuworld.com www.tuworld.com
www.tuworld.com www.tuworld.com 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, 4 4 5.6 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
www.tuworld.com Imgiry Prt www.tuworld.com.8.6.4. -. -.4 -.6 -.8 - - -.5.5.5 Rl Prt DSP-4 Z 8 of 84 Dr. Rvi Bill
DSP-4 Z 9 of 84 Dr. Rvi Bill Exmpl 4.6. Giv th pol-ro plot for H 9 8 7 6 5 4 9 8 7 6 5 4 Solutio From H 9 4 5 6 7 8 9 8 7 6 5 4 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 9 8 7 6 5 4 9 8 7 6 5 4 %Pol-ro plot b [: 9]; [, ]; pl b, -.5 - -.5.5.5 - -.5.5 9 Rl Prt Imgiry Prt www.tuworld.com www.tuworld.com
www.tuworld.com Imgiry Prt www.tuworld.com Exmpl 4.6.4 Giv th pol-ro plot for y x +.8 x.8 x.45 y Solutio Th ros r,.8 Th pols r giv by, ±.678.8 4.8.589 d.99.45]. For th ATLAB progrm th cofficit vctors r b [,.8, -.8] d [,, %Pol-ro plot b [,.8, -.8]; [,,.45]; pl b,.8.6.4. -. -.4 -.6 -.8 - -.5 - -.5.5 Rl Prt DSP-4 Z 4 of 84 Dr. Rvi Bill
www.tuworld.com www.tuworld.com 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
www.tuworld.com www.tuworld.com 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. + +5 + 9 + Q D + 6 + 4 4 9 + 6 5 6 5 5 + 9 9 7 + 8 7 8 DSP-4 Z 4 of 84 Dr. Rvi Bill
www.tuworld.com www.tuworld.com Thus X + + 5 + 9 + 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: + + 5 + 9 + Q D + 6 + 4 4 9 + 6 5 6 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 + + + 9/ / 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
www.tuworld.com www.tuworld.com 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 + + + 9/ / 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
www.tuworld.com www.tuworld.com + + + + 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
www.tuworld.com www.tuworld.com 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
www.tuworld.com www.tuworld.com 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 5 5... 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
www.tuworld.com gitud Phs - Rdis www.tuworld.com subplot,,, plotw, glh; xlbl'frqucy \omg', ylbl'phs - Rdis'; grid.5.5-8 -6-4 - 4 6 8 Frqucy 4 - -4-8 -6-4 - 4 6 8 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 46 4 5,.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
www.tuworld.com www.tuworld.com 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
www.tuworld.com Phs - Rdis gitud www.tuworld.com.5.5-4 - - - 4 Frqucy Rd.4. -. -.4-4 - - - 4 Frqucy Rd Exmpl 4.8.4 [] 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, 4 4 5.6 d.6 Uit circl Im ROC >.6 R.6.6 DSP-4 Z 5 of 84 Dr. Rvi Bill
www.tuworld.com www.tuworld.com 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.5.5... 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
www.tuworld.com Phs - Rdis gitud www.tuworld.com -4 - - - 4 Frqucy Rd - - -4 - - - 4 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
www.tuworld.com www.tuworld.com 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 +
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 www.tuworld.com www.tuworld.com
www.tuworld.com www.tuworld.com 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
www.tuworld.com www.tuworld.com Pluggig i th iitil coditios y 4 d y Y Y 4 Y 4 + Y 6 + 5 + / 4 Y.5.5 + / 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 5 5 5 y,,,,... 4 6 64 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,,,,... 4 8 4
www.tuworld.com www.tuworld.com 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.5.5.5.5.5 Y.5.5 Y.5.5.5 Th domitor o th right hd sid hs roots t, Y A B.5.5 4.5.5.9.89.5.89.5.9.9.89.55.45 A.9 DSP-4 Z 57 of 84 Dr. Rvi Bill 5.9 d.89 4 B +.89.55.45 Y.9.89 y.55.9.45.89, 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:
www.tuworld.com x y www.tuworld.com %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 [-.5.75 -.5.88 -.79.46 -.. -.5.] Iput Squc.5 -.5-5 5 5 Output Squc.4. -. -.4 5 5 5 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
www.tuworld.com www.tuworld.com 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