DIGITAL SIGNAL PROCESSING BEG 433 EC

Transcription

1 Dowloadd from DIGITAL SIGAL PROCESSIG BEG EC.Yar: IV Smstr: II Tachig Schdul Examiatio Schm Hours/W Thory Tutorial Practical Itral Assssmt Fial - / Thory Practical* Thory Practical** * Cotiuous ** Duratio: hours Cours obctivs: To rovid Total 5. Discrt sigals 5. Discrt sigals uit imuls, uit st, xotial squcs. Liarity, shift ivariac, causality. Covolutio summatio ad discrt systms, rsos to discrt iuts. Stability sum ad covrgc of owr sris.5 Samlig cotiuous sigals sctral rortis of samld sigals. Th discrt Fourir trasforms 5. Th discrt Fourir trasform DFT drivatio. Prortis of th DFT, DFT of o-riodic data. Itroductio of th fast fourir trasform FFT. Powr sctral dsity usig DFT/FFT algorithms. Z trasform 8. Dfiitio of Z trasform o sidd ad two sidd trasforms. Rgio of covrgc rlatioshi to causality. Ivrs Z trasform by log divisio, by artial fractio xasio.. Z trasform rortis dlay advac, covolutio, Parsval s thorm.5 Z trasfor trasfr fuctio H Z trasit ad stady stat siusoidal rsos ol ro rlatioshis, stability.6 Gral form of th liar, shift ivariat costat cofficit diffrc quatio.7 Z trasform of diffrc quatio.. Frqucy rsos. Stady stat siusoidal frqucy rsos drivd dirctly from th diffrc quatio. Pol ro diagrams ad frqucy rsos. Dsig of a otch filtr from th ol ro diagram. 5. Discrt filtrs 6 5. Discrt filtrs structurs, scod ordr sctios laddr filtrs frqucy rsos 5. Digital filtrs fiit rcisio imlmtatios of discrt filtrs 5. Scalig ad ois i digital filtrs, fiit quatid sigals quatiatio rror liar modls. - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ -

2 Dowloadd from 6. HR Filtr Dsig 7 6. Classical filtr dsig usig olyomial aroximatios Buttrworth Chbishv 6. HR filtr dsig by trasformatio matchd Z trasform imuls, ivariat trasform ad biliar trasformatio 6. Alicatio of th biliar trasformatio to HR low ass discrt filtr dsig 6. Sctral trasformatios, high ass, bad ass ad otch filtrs. 7. FIR Filtr Dsig 7. FIR filtr dsig by fourir aroximatio th comlx fourir sris 7. Gibbs homa i FIR filtr dsig aroximatios, alicatios of widow Fuctios to frqucy rsos smoothig rctagular haig Hammig ad Kaisr widows. 7. FIR filtr dsig by th frqucy samlig mthod 7. FIR filtr dsig usig th Rm xchag algorithm 8. Digital filtr Imlmtatio 8. Imlmtatios usig scial uros DSP rocssors, th Txas Istrumts TMS. 8. Bit srial arithmtic distributd arithmtic imlmtatios, ilid imlmtatios Laboratory:. Itroductio to digital sigals samlig rortis, aliasig, siml digital otch filtr bhaviour. Rsos of a rcursiv HR digital filtr comariso to idal uit saml ad frqucy rsos cofficit quatiatio ffcts.. Scalig dyamic rag ad ois bhaviour of a rcursiv digital filtr, obsrvatio of oliar fiit rcisio ffcts. Dowloadd from -

3 Dowloadd from Digital Sigal rocssig:- Alicatio:- Sch rcogitio. Tlcommuicatio. Digital Sigal rocssig ovr aalog sigal rocssig: Accuracy. Offli rocssig. Softwar cotrol. Char tha aalog coutrart. Basic lmt of D.S.P systm:- Aalog iut sigal A/D Covrtr Digital sigal rocssig D/A Covrtr Aalog outut sigal Digital iut sigal Digital outut sigal Fig: Bloc diagram of DSP - Most of th sigal coutrig scic ad girig ar aalog i atur i. th sigals ar fuctio of cotiuous variabl substac i usually ta o valu i a cotiuous rag. - To rform th sigal rocssig digitally, thr is d for itrfac btw th aalog sigal ad digital rocssor. This itrfac is calld aalog to digital covrtr. Th o/ of A/D covrtr is digital sigal i. aroriat as a i/ to th digital rocssor. - Digital sigal rocssor may b a larg rogrammabl digital comutr or small microrocssor rogram to rform th dsird oratio o i/ sigal. - It may a also b a hardwird digital rocssor cofigur to rform a scifid st of oratio o th i/ sigal. - Programmig machi rovid th flxibility to chag th sigal rocssig oratio through a chag i softwar whras hardwird m/c ar difficult to rcofigur. - I alicatio whr th distac o/ from digital sigal rocssor is to b giv to th usr i aalog form, w must rovid aothr itrfac o th digital domai ito aalog domai. Such a itrfac is calld D/A covrtr. Advatag of Digital ovr aalog sigal rocssig:-. A Digital rogrammabl systm allows flxibility i rcofigurig th digital sigal rocssig oratio simly by chagig th rogram. Rcofiguratio of aalog systm usually imlis rdsig of hardwar followd by tstig ad vrificatio to s that if orats rorly.. Digital systm rovid much bttr cotrol of accuracy rquirmts.. Digital systm ar asily stord o magtic mdia without loss of sigal byod that itroduc i A/D covrsio. As a cosquc, th sigals bcom trasortabl ad ca b rocssd offli i a rmot laboratory. - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ -

4 Dowloadd from Digital sigal rocssig mthod also allows for th imlmtatio of mor sohisticatd. 5. I som cass a digital imlmtatio of sigal rocssig systm is char tha its aalog coutr art. * Sigal: It is dfid as ay hysical quatity which is a fuctio of o or mor iddt variabl ad cotais som iformatio. I lctrical ss, th sigal ca b voltag or currt. Th voltag or currt is a fuctio of tim as a iddt variabl. Th iddt variabl i th mathmatical rrstatio of a sigal may b ithr cotius or discrt. Cotius tim sigals ar dfid aalog cotius tims. Cotious tim sigals ar oft rfrrd to as aalog sigals. Discrt tim sigals ar dfid as crtai tim istat. Digital sigals ar thos for which both tim ad amlitud discrt. Discrt- tim sigals or squc:- D.T.S ar r. mathmatically as a squc of umbrs. A squc of umbrs x i which th th o i th squc is dotd by x ad writt as: x { x} - ifiity < < ifiity Figur: uit saml or uit imuls squc: It is dotd by δ ad dfid as: δ - - * Uit st squc: It is dotd by U ad is dfid as: U,, < U δ δ- δ- Dowloadd from -

5 Dowloadd from δ U δ i. th valu of uit st squc at tim is qual to th accumulatd sum of valu at idx ad all rvious valu of imuls squc. Covrsly th imuls squc ca b xrssd as th first bacward diffrc of uit st squcs. i.. δ u u- * Uit ram squc:- It is dotd by u r ad dfid as u r < * Exotial Squc:- Th xotial sigal is a squc of th form x a for all. If th aramtr a is ral, th x is ral sigal. Fig illustrat x for various valus of aramtr a. <a< a> a< < a < : Eg a ½ i. /, ½, ¼, /8 xotial dcrasig s figur i Fig: Grahical rrstatio of Exotial sigals. - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 5

6 Dowloadd from a >, E.g, a i.,,, 8 Exotial icrasig S figur ii - < a < Eg.: -/ i. -/, -/, ¼, -/8 S fig iii a <- E.g a - -, -,, -8 S figuriv Exotial squc:- Wh th aramtr a is comlx valus, it ca b xrssd as: a r θ Whr r ad θ ar w aramtrs. Hc, w ca xrss x as: r θ r cos θ si θ Sic, x is ow comlx valus, it ca rrstd grahically by lottig th ral art, x r cos θ as a fuctio of ad saratly lottig th imagiary art. x i rsiθ as a fuctio of. Fig. illustrats th grahs fo ad x i. Dowloadd from -6

7 Dowloadd from W obsrv that th sigals x ad x i ar damagd dcayig xotially, i. r <r cosi fuctio ad damd si fuctio. If r, th damig disaars ad x, x i ad x hav fixd amlitud which is uity. Altrativly, th sigal x ca b rrstd grahically by th amlitud fuctio. x A r Ad has fuctio x ø θ Rrstatio of discrt-tim sigal: - Fuctioal rrstatio:- E.g x, for,, for lswhr, Tabular rrstatio: Squc rrstatio: x {,,,,,,.} Ifiit duratio. {,,, fiit duratio - oit squc Grahical rrstatio:- Figur: Dat: 66/5/ Liarity: A systm is calld lir of surositio ricial alis to that systm. This mas that th lir systm may b dfid as o whos rsos to th sum of wightd iuts is sam as th sum of wightd rsos. Lt us cosidr a systm. If x is th iut ad y is th outut. Similarly y is th rsos to x. Th for lir systm. a x a x a y a y.. For ay oliar systm th ricil of surositio dosot hold tru ad quatio i is ot satisfid. umrical: For th followig systm, dtrmi whthr th systm is lir or ot. y x Solutio: y x y x - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 7

8 Dowloadd from Alyig surositio ricil suos, x a x a x Th, y [a x a x ] a x a x y a y a y a [x ] a [x ] x a a x a a Sic, y y. Th systm is oliar. ii y x Solutio: y x x y Alyig iut such that, x a x a x Th rsos will b, y [a x a x ] ad agai, y a y a y a x a y y y Th systm is oliar. Shift ivariac:- A systm is shift ivariat, if th iut outut rlatioshi dosot vary with shift. I othr words for a shift ivariat systm shift i th iut sigal rsults i corrsodig shift i outut. Mathmatically, x y Which mas that y is th rsos for x. If x is shiftd by, th outut y will also b shiftd by sam shift i. x y Whr is a itgr. If th systm dosot satisfy abov xrssio, th th systm is calld shift variat systm. Th systm shiftig both liarly ad tim ivariat rortis ar oularly ow as lir tim ivariat systm or simly LTI systms. umrical: Chc whthr th systm ar shift ivariat or ot. i y x Solutio: Lt us shift i iut by, th th outut will b, Dowloadd from -8

9 y x y y Hc th systm is shift ivariat systm. ii y xm Lt us shift i iut by th, rsos will b, y xm ad th shift i outut by will yilds. y x[ m m ] Th systm is shift variat. Dowloadd from Causality:- A systm is causal of th rsos dos ot bgi bfor th iut fuctio is alid. This mas that of iut is alid at, th for causal systm outut will dd o valus of iut x for. Mathmatically, y T[ x, ]..i Th rsos of causal systm to a iut dosot dd o futur valus of that iut but dds oly o rst or ast valus of iut. O th othr hads, of th rsos of th systm to a iut dds o futur valus of th iut, th systm is ocausal. A o causal systm dosot satisfy quatio i. Causal systm ar hysically raliabl whras o causal systm caot b imlmtd ractically. Thr is o systm ossibl ractically which ca roduc its outut bfor iut is alid. Th quatio, y x dscribs th causal systm ad y x x dscrib th o causal systm. Mmory lss systm:- A systm is rfrrd to as mmory lss if th o/ y at vry valu of dds oly o th i/ x at th sam valu of. * Rsos of LTI systm to Arbitrary iut covolutio solutio:- Cosidr ay arbitrary discrt tim sigal x[] as show i fig: Dat: 66/5/5 - A discrt tim squc sigal b rrstd by a squc of idividual imulss as show i figur: - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 9

10 Dowloadd from xδ xδ W ca writ, x-δ x δ x δ- x δ-.. x δ... i covolutio sum Suos h is th o/ of LTI systm wh δ is i/. Thrfor, th o/ for i/ x- δ is x- h. Th th o/ y for th i/ x giv i quatio i will b, y x h... ii Symbolically, y x* h iii y h*x.iv umricals: * Th imuls rsos of ivalid tim rsos is: h {,,, -} x {,,, } Solutio: Th rsos of th LTI systm is, y x h For, y x h Dowloadd from -

11 Dowloadd from - x h x h x h - For - y- x h -- x For - - Y- x h - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ -

12 Dowloadd from Agai tstig for v sid: For Y x h 8 x- - For - Y x h xh- For 6 Y {.,,, 8, 8,,-, -,. Figur: * Dtrmi th o/ y of rlaxd liar tim ivariat systm with imuls rsos: h a u, u < Wh th i/ is uit st squc i; x u Solutio:- y x h For, y x h Dowloadd from -

13 Dowloadd from a a a a a h- For : y x h a a a a a h- xh- - For : y x h aa a a xh- a r S r For > y a.a a a For < y a y lim y lim a a - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ -

14 Dowloadd from y -a a aa * Itrcoctio of LTI systm:- h h y h*h y Wh two LTI systms with imuls rsos h ad h ar i cascad form th ovrall imuls rsos for th cascadd imuls systm will b, h h * h i * Th aralll combiatios of LTI systms ad quivalt systm is show blow: h x h y hh y Dtrmi th imuls rsos for th cascad of two LTI systms havig imuls rsoss. h / u h /u u Solutio:- Th ovrall imuls rsos is hu h * h h h / / h / /6 h Dowloadd from -

15 Dowloadd from /6 / h- /6 / h- > / / / / / / / / - / [ / ], ot:- If w hav L LTI systm is cascad with imuls rsoss h ad h.h L, th imuls rsos of quivalt LTI systm is h h * h *h..*h L Discrt systm rsos to discrt iut:- For a discrt-tim systm, cosidr as iut squc x w, for - << th outut of LTI systm with imuls rsos h is, y h x h w w h If w dfi, w H w w h Th, y H w w H w dscribs th chag i comlx amlitud of comlx xotial as a fuctio of frqucy w. H w is calld frqucy rsos of th systm. I gral H w is comlx ad ca b xrssd i trms of its ral ad imagiary arts as: H w H R w H I w Or, I trms of magitud ad has as, - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 5

16 H w H w Hw Lt, x Acosw o ø A/ wo. ø A/ -wo. -ø Th rsos to x A/ ø wo is, y H w A/ ø wo Th rsos to x A/ -ø -wo is, y H -w A/ -ø. - wo Thus, total rsos is, y A/ [H wo -ø. wo H -wo -ø. -wo ] A H -wo cosw o øθ. Whr, θ H wo Dowloadd from Dat: 66/5/ Stability: W dfid arbitrary rlaxd systm as BIBO stabl. If a oly if o/ squc y is boudd for vry boudd i/ x. If x is boudd thir xist a costat M x such that, x M < x Similarly if o/ is boudd thir xists a costat M y such that y M y < for all, ow giv such a boudd iut squc x to LTI systm with covolutio formula. y h x h x h x x M h From this xrssio w obsrv that th systm is boudd if th imuls rsos of th systm satisfid coditio, S h < That is liar tim ivariat systm is stabl if its imuls rsos is absolutly summabl. # Dtrmi th rag of valu of th aramtr a for LTI systm with imuls rsos. h a u is stabl. Solutio, Dowloadd from -6

17 Dowloadd from S h < h a a a a a a... Providd that a < Thrfor th sytm is stabl if a < othrwis it is ustabl. # Dtrmi rag of valu of a,b for which LTI systm with imuls rsos. h a S b h < b is stabl. a Th first sum covrag for a <. Th scod sum ca b maiulatd as, b b b b b b b b β β β... β / β Providd that, β < or b > Hc th systm is stabl is both a < or b > Dat: 66/7/ # Dtrmi whthr th giv systm is BIBO stabl or ot. y / [ x x- x -] Solutio:- Assum that, x < M < for all. Th y / [ x x- x -] / [ x x- x - ] / [ M x M x M x ] M x BIBO Boudd iut boudd outut. - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 7

18 Dowloadd from Sic, M x is fiit valu, y also fiit. Hc th systm is BIBO stabl. # # Dtrmi whthr th giv systm is BIBO stabl or ot. Y r x whr r>. Solutio: Assum y r x r x With r >, th multilyig factor r divrgs for icrasig ad hc o/ ot boudd. Hc th systm is BIBO ustabl. yquist Samlig thorm:- A sigal whos sctrum is bad limitd to B H Gw for w > B ca b rcostructd xactly form its samls ta uiformly at th rat R > BH saml/sc. I othr words, miimum samlig frqucy f s Bh. Cosidr a sigal gt whos sctrum is bad limitd to Bh. Samlig gt at th rat of f s h ca b accomlishd by multilyig gt by imuls δ Ts t cosistig of uit imulss ratig riodically vry T s scod. Whr, T s /f s Figur; Trigoomtric fourir sris of imuls trai, δ Ts t cos wst cos wst coswst... T g t g t δ t s [ ] Ts Whr, W s /T s s s st Ts [ g t g tcos w t g tcos w t g tcosw...] Usig modulatio rorty, gt cosw s t F.T Gw-w s Gww s G w s s T T s s [ G w G w w G w w...] G w w s If w wat to rcostruct gt from gt bar w should b abl to rcovrd Gw form G w. This is ossibl if thr is o ovrla btw succssiv cycl of G w gratr th B.. Figur shows that this rquirs f s Dowloadd from -8

19 Dowloadd from From figur w s that gt ca b rcovrd form saml g t by assig samld sigal through idal low ass filtr with badwidth B h. Samlig of aalog sigals: Dat:66/7/6 Thr ar may ways to saml aalog sigal. W limit our discussio to riodic or uiform samlig which is h tys of samlig usd most oft i toic i ractic. This is dscribd by th rlatio. x a T, - < < Whr x is discrt-tim sigal obtaid by talig samls of aalog sigal x t vry T scod. T samlig riod or saml itrval. F s /T damig frqucy or samlig rat saml/sc or H t T /F s I gral th smalig of cotiuous tim siusoidal sigal t Acos f t θ x a With samlig rat f s rsults discrt tim sigal. T x Acos ft θ f Whr, f Rlativ frqucy of siusoid. f s Q. Cosidr th aalog sigal x a t cost a Dtrmi th miimum salig rat rquir to avoid aliasig. b Suos that th sigal is samld at th rat F s h What is th discrt tim sigal obtaid aftr samlig. c Suos that th sigal is saml at th rat F s 75 h what is th discrt tim sigal obtaid aftr samlig. d What is th frqucy < f< F s / of a siusoids that yilds samls idtical to thos obtaid i art c. Solutio:- t cos t f F 5 h b Miium samlig rat h c F s h d F s 75 h cos /f s cos /75 cos cos / cos -/ cos / F s 75 h, f / f F /F s, F f F s 5 h - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 9

20 Dowloadd from yt cos5 t Q. Cosidr th aalog sigal at x a t cos5 t si t-cos t. What is th qust rat for this sigal. Solutio: - Frqucy rst i th sigal, F 5h, F 5 h, F 5 h F max 5 h yqust rat F max h. Q. Cosidr th aalog sigal at x a t cos t5si6 tcos t. a What is th yqust rat for th sigal. b Assum ow that w saml th sigal usig samlig rat F s 5 samls r scod. What is th discrt tim sigal obtaid aftr salig. Solutio: a F h F h F 6h yqust rat h b F s 5 h 5h x cos /F s 5si6 /F s cos /F s cos /5 5si6 /5 cos /5 cos /5 5si /5cos 6/5 cos /5 5si /5cos /5 cos /5-5si /5cos /5 cos/5-5si/5 Chatr:- Discrt Fourir trasform:- Frqucy domai samlig: Discrt Fourir trasform. DFT Lt us cosidr ariodic discrt-tim sigal x with Fourir trasform w w x.i Dat: 66/7/6 Suos that w saml w riodically i frqucy at a sacig of δw radia btw th succssiv samls. Sic w is riodic with riod oly samls i th fudamtal frqucy rag ar cssary. W ta quidistat saml i th itrval < w <. With saml sacig δw / ω δω Fig: Frq. domai samlig. ω Dowloadd from -

21 Dowloadd from - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - If w valuat, quatio i w gt, W /. K, to - x / Summig i quatio ca b subdividd ito ifiit umbr of summatios whr ach sum cotais trms Thus, / / / x x x l / l l x If w chag th idx i th ir summatio form to -l ad itgratig th ordr of summatio w obtaid, / l x.. Th sigal l l x.. Obtaid by riod rrstatio of x vry samls is clarly riod with fudamtal riod. Sic x is riod xtsio of x giv by quatio it is clar that x ca b rcovrd from x if thr is o alisig i th tim domai that is x is limitd to lss tha th riod of x L x x L > L

22 Dowloadd from x < L I summary a fial duratio squc x of a lgth L as fourir trasform w for < w <.5 Wh w saml w at qual sac frqucy w K/,,, - Whr, L, Th rsultat saml ar, K / x. 6 Whr,,, - L x Th rlatio i quatio 6 is a formula for trasformig squc x of lgth L < ito a squc of frqucy samls {} of lgth L. Sic th frqucy samls ar obtaid by valuatig th fourir trasform w at a st of qually sacd discrt frqucis. Th rlatio i quatio 6 is calld discrt fourir trasform DFT of x. w x ca b writt as, x / a,, - With Fourir cofficit,.7 a / / x,,..-.8 Form quatio ad 8 a /,,..-.9 Thrfor, x / /,,..-. This rlatio allows us to rcovr th squc x from frqucy saml. x / /,.-. This is calld ivrs DFT IDFT Dat:- 66/7/7 DFT as liar trasformatio:- Dowloadd from -

23 Dowloadd from Th formula for DFT ad IDFT is giv by x,,.-.. w x w,,.-.. Whr W -/. Lt us dfi -oit vctor x of sigal squc x,,, - ad -oit vctor of frqucy samls ad * matrix W as x x x W W W W. W x W W W W W With thr dfiitio -oit DFT may b xrssd i matrix W x.. Whr, W matrix of lir trasformatio. Equatio ca b ivrtd by r multilyig both sid by W -. Thus w ca obtaid x w -..5 Which is xrssio for IDFT x / W x..6 Whr, W * comlx cougat of matrix W W - W * /..7 Q. Comut DFT of -oit squc x {,,, } Solutio:- x / - -. x x x / x x x x 6 / x x / x x / 6 By matrix mthod, - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ -

24 Dowloadd from Dowloadd from - W x W * x * 6 Q. Comut IDFT for th frqucy comot { 6,, -, } Solutio:- x W * / x W * / 6 W x W * / ¼ 6

25 Dowloadd from ¼ ¼ Prortis of DFT:- Th otatio usd to dot -oit DFT air x ad as x DFT Priodicity: If x DFT Whr, x x for all. for all. Proof: x / / K x / / / / Liarity: - If x DFT ad x DFT Th, for ay ral or comlx valud costats a ad a x a x a x DFT a a Proof:- a x a x DFT x / a x a a a a x a x / / Circular Symmtris of squc: Dat:66/8/ - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 5

26 Dowloadd from -oit DFT of fiit duratio squc x of lgth L is quivalt to -oit DFT of riodic squc x of riod which is obtaid by riodically xtdig x i. l x x l l x x x l Fiit duratio squc, x x < <- othrwis Is rlatd to origial squc by x by circular shift. This rlatioshi is show grahically as follows. This rlatioshi is show i figur x x x Dowloadd from -6

27 Dowloadd from x x x x x x x x x x I gral th circular shift of th squc is rrstd as idx modulo. i. x x- For xaml, K, That imlis, x x- x x- x x x x Hc x is shiftd circularly by uits i tim. whr th coutr cloc wis dirctio is slctd as th v dirctio. Thus w coclud that circular shift of - oit squc is quivalt to liar shift of its riod xtsio ad vis vrsa. Symmtry rortis of DFT:- Lt us assum that -oit squc x ad its DFT ar both comlx valud. Th th squc ca b rsrstd as x x R x I < <-.. R I <<-.. x / x x x R R R I R x I Similary, x R I cos / x R,, - cos / I si / si / x cos / x si /.. I si / x cos / I R si / x I cos / - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 7

28 x R / x I / Ral valud sigal:- If x is ral, - * - Proof:- - R R x x Dowloadd from cos / si / 5 I si / cos /.6 / / - * Ral ad v squc:- x x- Tha, I, DFT rducs to x cos / < <- IDFT Rducs to, x / cos / <<- Ral ad odd squc:- If x x- <<- Tha, R, DFT rducs to, - x si /, << - IDFT rducs to, x / si / < < - Purly Imagiary squc:- x x I R x si / I I xi cos / If x I is odd, th I ad hc is urly ral. If x I is v th, R ad hc is urly imagiary. I Dowloadd from -8

29 Dowloadd from - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 9 Multilicatio of two DFTs ad circular covolutio:- Suort w hav two fiit duriaot squcs of lgth, x, x. Thir rsctiv DFTs ar, / x,, -.. / x,,.-. If w multily two DFTs togthr th rsult i DFT say of a squc x of lgth. Lt us dtrmi th rlatioshi btw x ad th squc x ad x. W hav x / / m x / / m / / / / l l l x x / / l m l l x x ow,, a a a a a a Whr, a m--l/ othrwis gr is m m l a, it Hc, x m / m x x. m,,..- x m m x x

30 Dowloadd from Th xrssio has th form of covolutio sum. Th covolutio sum ivolvs idx m- is calld circular covolutio. Thus w coclud that multilicatio of DFT of two squcs is quivalt to th circular covolutio of two squcs i tim domai. Q. rform th circular covolutio of th followig two squcs. x {,,, } x {,,, } Solutio, x m, x x m x m x x m x x x x x x x x x x x x x x - x - 6 x x x 6 x x - 6 x - 8 x x x Dowloadd from -

31 Dowloadd from - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - x - 6 x x - x 6 x x x - 8 x x - 6 x {, 6,, 6} Solutio by DFT ad IDFT mthod: * 6 6 IDFT: x W * / *

32 Dowloadd from Dowloadd from - ¼ 6 Parsval s thorm:- For comlx valud squc x ad y x DFT y DFT Y Th, * * Y y x Proof:- * xy r y x Circular cross corrlatio squc. / l xy xy R l r * xy Y r I Scial cas, y x * x y x Which xrsss th rgy is fiit duratio squc x i trm of frqucy comot {} Dat: 66/8/ Fast Fourir Trasform FFT:- / x Th comlx multilicatio i dirct comutatio of DFT is ad by FFT comlx multilicatio i / log. Wh umbr of oits is qual to, th comlx multilicatio i dirctio comutatio of DFT is 6 ad for FFT its valu is. Hc th icrmt factor is. / * l xy Y l r

33 Radix- FFT algorithm:- Dowloadd from If x b th discrt tim squc th its DFT is giv by x w,,.- Divid -oit data squc ito two / data squc f ad f corrsodig to th v umbr ad odd umbr samls of x rsctivly. f x f x,, /- Thus f ad f ar obtaid by dcimatig x by a factor of ad hc th rsultig FFT algorithm is calld dcimatio i tim algorithm. But v / m x w xm w odd x w / m xm w / / / W / W / / m f m w / W f m m m W F w m m / F Whr F ad F ar / oit DFT of squcs f m ad f m rsctivly. x x x x6 / oit DFT F F F F x x x5 x7 / oit DFT F F F F Havig rformd that DIT oc, w ca rat th rocssor for ach of squc f ad f. Thus f would rsult i two / oit squc. v f v f,,, /- Ad f would rsult. v f v f,,,./ - Th - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ -

34 Dowloadd from F v v w / x x x W W W W W W W W 6 W W W W x W W W x5 x6 x7 W W W W W W 6 Fig: 8-oit DIT FFT algorithm W 5 W 6 W m m m m m q m q W r/ m m W r m q m q m W r/ m q m - W r m q [W / -] m q W r - m q Fig: Basic buttrfly comutatio i DIT FFT algorithm. # Comut 8 oit DFT for th squc x {,,,,5,6 }. Usig DITFFT algorithm or DIF FFT algorithm. x x x x W W W W x - W x5 - W W 5 x6 - W W 6 x7 - W W 7 Dowloadd from -

35 Dowloadd from Q. Comut -oit DFT for th squc x {, 6,, 6} usig FFT algorithm. Dat:66/8/ x x x6 W - x6 W 6 W Q. Comut oit DFT for th squc x {, 6,, 6 } usig FFT algorithm. x x x - x - 8 W W W W 6 - Chatr: - Z-trasform:- Th -trasform of a discrt-tim sigal x is dfid as th owr sris. x.. Wh is comlx variabl. Dat: 66/8/ - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 5

36 Dowloadd from Th ivrs rocdur i. obtaiig x from is calld ivrs -trasform. -traasform of a sigal x is dotd by Z{x}. Whr as th rlatioshi btw x ad is idicatd by x.. ROC Rgio of covrgc: ROC of is th st of all valus of for which attis a fiit valus. # Dtrmi -trasform of followig fiit duratio sigals. x {,, 5, 7,, } x ROC: tir -la xct. x {,, 5, 7,, } x 5 7 ROC: Etir -la. Exct ad x δ x [ δ ] ROC: Etir -la. x δ-u > x [ δ ] ROC: Etir -la xct 5 x 5 δ > 5 x5 δ [ ] ROC: Etir la xct. ROC rlatioshi to casusality:- Dowloadd from -6

37 Dowloadd from Lt us xrss comlx -variabl i olar form as. r θ.. Whr r θ agl. Th ca b xrssd as, rθ I th ROC of. But, θ x r. < x r θ x r.. Hc is fiit of th squc x r is absolutly summabl. Th roblm of fidig ROC for x is quivalt to dtrmiig th rag of valus of r for which th squc x r - is absolutly sum abl. x r x r x x r. r If x covrgs i som rgio of comlx lai both summatio i quatio must b fiit i that rgio. If th first sum i quatio covrgs thir must xists valus of r small ough such that th roduct squc x- r, < < is absolutly summabl. Thrfor ROC for th first sum cosists of all oits i a circl of som radius r whr r < as illustratd i th figur. I m -la R Dat: 66/8/5 ow if th d trm i quatio covrgs thr must xists valus of r larg ough such that x roduct squc,.. hc ROC for scod sum i quatio cosists of all oits outsid a r circl of radius r >r as show i fig. - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 7

38 Dowloadd from I m r R Sic th covrgc of rquirs that both sums i quatio b fiit it follows that ROC of is grally scifid as th aual rgio i th la r <r<r which is commo rgio whr both sums ar fiit. Which is show i figur. I m r r R If r >r thr is o commo rgio of covrgc for th two sums ad hc dos ot xist. I m r r R umrical: # Dtrmi -trasform of th sigal. α u α, >, < Solutio: Dowloadd from -8

39 Dowloadd from x α α α α... α for α < ROC: > α I m x r α R # Dtrmi -trasform of th sigal. - α u-- -α, < -, > Solutio:- x α l l α l l l l α [ α α...] α - < for α α ROC : < α α - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 9

40 Dowloadd from x I m r -la R # Dtrmi -trasform of th sigal. α u b u-- Solutio: x b α l b α l Th first owr sris covrgs if b - < i. < b ad scod owr sris covrgs if α - < i, > α Cas I: b < α I m -la R Cas II: b > α Dowloadd from -

41 Dowloadd from I m -la α R b α b α α b αb ROC: α < < b Dat: 66/8/ liarity : If, x, x Th x a x a x a a Proof: x a x a x a x a x a a # Dtrmi -trasform ad ROC of th sigal x[] [ - ] Proof: Lt, x u x u Th, x x -x Accordig to liarity rorty. - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ -

42 Dowloadd from - Z {x } ROC: > Z {x } ROC: > Th itrsctio of ROC of ad is >. ow > # Dtrmi -trasform of th sigals: a x cosw o u b x siw o u Solutio: x cosw o u w w / u w w u u Usig liarity rorty. w { w } { Z u Z u } w Z{ u } ROC: > w w Z{ u } ROC: > w.. w w Cosw ROC: > cos w w b x w u Siw cos w ROC: > Tim Shiftig:- If x Th, x- - Proof: x- x Put - m Dowloadd from -

43 Dowloadd from x m m m x m m - m # By alyig th tim-shiftig rorty dtrmi -trasform of x ad x form -trasform of x giv as, 5 x {,, 5,7,, }, x {,,, 7,, } x {,,,,5,7,,} Solutio: x, x x - ow, ROC: Etir -la xct ad Roc: tir -la xct. # Dtrmi th -trasform of th sigal. x othrwis Solutio:.... if if Altrativ mthod:- u u- Usig liarity ad tim shiftig rorty. Z{u} Z{u-} ROC: > xt mthod - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ -

44 Dowloadd from u u- - Dat:66/8/5 Scalig i -domai:- If x ROC, r < <r Th a - x a - ROC: a r < < a r For ay ral costat a ral or comlx. Z[ a x ] a x x a ROC : r < a - <r a r < < a r a # Dtrmi -trasform of th sigals. a x a cosw o u b x a siw o u Solutio: x a cosw o u x cosw o u cos w Z [cos wu ] ROC: > cos w Th, a x a cos w Z[ a x ] a ROC: > a a cos w a b x a siw o u a si w a cos w a ROC: > a Tim rvrsal: If x ROC, r < <r Th x- - ROC: /r < </r Proof: Dowloadd from -

45 Dowloadd from Z [{ x }] x Put, l - l x l l x l l - l ROC of - is r < <r or, /r < </r ot that ROC of x is th ivrs of that for x- # Dtrmi th -trasform of th sigal. x u- Solutio: u > Usig tim rvrsal rorty. u- < 5 Diffrtiatio i - trasform If x d Th x - d Proof:- x By diffrtiatio, d x d { x Z{ x } d Z{ x } ot that both trasform hav sam ROC. d # Dtrmi -trasform of th sigal. x a u Proof: x a u - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 5

46 Dowloadd from x a a ROC; > a x x d Z{ x } d a a a a ROC: > a # Dtrmi th sigal x whos -trasform is giv by loga -, > a Solutio: d. a d a d a d a. a a Taig ivrs -trasform. x a-a - u- x - -. a / u- Dat: 66/8/6 If x x Th x.x Th ROC of is at last th itrsctio of th for ad. Proof: Th covolutio of x ad x is dfid as x x x -trasform of x as, x Dowloadd from -6

47 x x x x x x x Dowloadd from # Comut covolutio x of th sigals {,-, } 5 x othrwis Ivrs -trasform. x.i Multlyig both sids of i by - ad itgrat both sids ovr a closd cotour withi th ROC of which closd th origi. ow, d x d... C Whr C dots closd coutr i th ROC of. W ca itrchagd th ordr of itgratio ad summatio o right had sid of. d x d... From Cauchy itgral thorm, C d C Th, Figur: d x x d.. C 7 Multilicatio of two squcs:- - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 7

48 Dowloadd from Dowloadd from -8 If x x Th x x x C dv v v u Whr C is sth closd coutr that closs th origi ad lis withi th ROC of commo to both v ad /v x x x Whr, C dv v v x C dv v v v x * * C dv v v x v 8 Parsval s thorm:- If x ad x ar comlx valud squc, th, C dv v v v x x * * / * c c for all. Whr is ratioal, th xasio ca b rformd by log divisio. # Dtrmi ivrs -trasform of.5.5 Wh a ROC: > b ROC : <.5

49 Dowloadd from Solutio: a ROC: >, x is causal sigal. { } x {,.5, -.5, -.75,..} b ROC : <.5 x is aticausal sigal x {... 6,,,, } Iitial valu thorm: lim Fial valu thorm:- Lim x lim - # Th imuls rsos of rlaxd LTI systm is h α u α <. Dtrmi th st rsos of th systm whr tds to ifiity. Solutio:- y x*h x u h α u Y H α - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 9

50 Dowloadd from Dowloadd from -5 ow, Lim y lim -Y lim tds to - α α Systm fuctio trasfr fuctio of LTI Systm:- Th o/ of LTI systm to a iut squc x ca b obtaid by comarig th covolutio of x with uit saml of th systm. i. y x*h W ca xrss this rlatioshi i -domai as Y H Wh, Y -trasform of y -trasform of x H -trasform of h ow, H Y h H H rrst th -domai charactristics of th systm whr as h is corrsodig tim domai charactristics of th systm. Th trasform H is calld th systm fuctio or trasfr fuctio. Gral form of lir costat cofficit diffrc quatio:- Th systm is dscribd by lir costat cofficit diffrc form. M x b y a y. Taig trasform o both sids, M b Y a Y M b a Y M a b Y H. Thrfor a LTI systm dscribd by costat cofficit diffrc quatio has a ratioal systm fuctio from this gral form w obtaid two imortat scial forms:. if a, for < <. Equatio rducs to M b H all ro systm such a systm has fiit duratio imuls rsos ad it is calld FIR systm or movig avrag MA systm.

51 Dowloadd from - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 5. O th othr had if b, for < < th systm fuctio rducs to a b H. all ol systm Du th rsc of ols th imuls rsos of such a systm is ifiit i duratio ad hc it is IIR systm. # Dtrmi th systm fuctio ad uit saml rsos of th systm dscrib by th diffrc quatio y ½ y- x. Solutio: Taig -trasform o both sids, / Y Y / Y / Y H By ivrsio, h / u Dat: 66/9/6 Rsos of ol ro systm with o-ro iitial coditio:- Th diffrc quatio, M x b y a y i Suos that th sigal x is alid to th ol ro systm at. Thus th sigal x is assum to b a causal. Th ffcts of all rvious iut sigal to th systm ar rflctd i th iitial coditio y-, y-..y-. Sic th iut x is causal ad sic w ar itrstd i trmiatig th o/ sigal y for w ca us o sidd -trasform which allows us to dal with iitial coditio. ow, m b y Y a Y.. Sic x is causal, W ca st M a y a a b Y. A H o.. Whr, y a.

52 Dowloadd from From th o/ of th systm ca b sub dividd ito two arts. Th st art is ro stat rsos of th systm dfid i -domai as Y s H.. 5. Th scod comot corrsodig to o/ rsultig form iitial coditio. This o/ is ro iut rsos of th systm which is dfid i -domai as Y i 6 A Sic th total rsos is th sum of th two o/ comot which ca b xrssd i tim domai by dtrmiig ivrs -trasform of Ys Yi. Sratly ad addig th rsult. y y s y i..7 # Dtrmi th uit st rsos of th systm dscribd by th diffrc quatio y.9y-.8 y- x. Udr th followig iitial coditio. a y- y-. b y- y-. Taig o-sid -trasfrom. Y.9 Y y.8 Y For y- y- Y.9 Y.8 Y Y x.9 Ys.9 / [ ] [ y y ] / /.9 / *.9 /. 98 * 98 o ,. o.98 Y.9 By ivrsio, y s / / 5.5 / 5.5 / s [ ] y [ cos / 5.5 ] u b y- y-. o Yi A o a y Dowloadd from -5

53 Dowloadd from - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 5 o y a [ ] y y a y a [ ] Y i Y Y Y i s / / [ ] 8 / cos u y Dat: 66/9/ # Dtrmi wll ow fibaacci squc of itgr umbrs is obtaid by comutig ach trm as th sum of two rvious os, th first fw trms of th squcs ar,,,, 5, 8.. Dtrmi a clos form xrssio for th th trm of Fiboacci sris. Solutio:- Lt y b th th trm. Th, Y y- y- i With iitial coditio, y y-y- y yy- y- -, y-. Taig o sidd -trasform o both sids of i. y Y Y y Y Y Y Y Y Y Y Y Taig ivrs -trasformatio.

54 Dowloadd from Dowloadd from u y # Dtrmi st rsos of th systm y α y- x - < α < Wh iitial coditio is y-. Solutio: Taig o sidd -trasform, [ ] y Y Y α [ ] u Z Or, Y Y α α Or, Y α α Y α α α Y α α α α α Taig ivrsio,. u u u y α α α αα α α α α u y α α # Dtrmi th rsos of th systm y 5/6. y- /6 y- x to th iut sigal x δ δ Solutio: Taig -trasform i both sids Y Y Y Y Y Y

55 Y Y Taig ivrs -trasform, Dowloadd from y u Causality ad stability:- A causal LTF systm is o whr uit saml rsos u satisfis th coditio h, for <. W hav also show that ROC of -trasform of a causal squc is xtrior of a circl. A cssary ad sufficit coditio for a LTI systm to b BIBO stabl is h <. I tur this coditio imlis that H must cotai th uit circl withi it s ROC. Idd, Sic H h H h Wh valuatig o th uit circl i. H h Hc if th systm is BIBO, th uit circl is cotaid i th ROC of H. # A LTI systm is charactrid by th systm fuctio. H.5.5 Scify th ROC of H ad dtrmi h for th followig coditio. a Th systm is stabl. b Th systm is causal. c Th systm is aticausal. Solutio:- H Th systm has ols at ½ ad. - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 55

56 Dowloadd from Dowloadd from a Sic th systm is stabl, its ROC must iclud th uit circl ad hc it is ½ < <. Cosqutly h is o-causal.. u u h Th systm is ustabl. b Sic th systm is causal, its ROC is >. I this cas u u h Th systm is ustabl. c Sic th systm is aticausal it s ROC is </. I this cas,. u u h. Th systm is ustabl. Dat: 66/9/ Trasit ad stady stat rsos: # Dtrmi th trasit ad stady stat rsos of th systm charactrid by th diffrc quatio y.5y- x. Wh th iut sigal is x cos u. Th systm is iitially at a rst i. it is rlaxd. Solutio: Taig -trasform,.5 Y Y.5 Y cos cos cos w w u w Z o o o cos cos..5 Y.5 / / / 8.7 /

57 Dowloadd from Th atural or trasit rsos is yr 6..5 u ad forcd or stady rsos is 8.7 / 8.7 / y u fr [ ].56 cos 8.7 u Pol-ro diagram:- H I m R Fig: ol-ro diagram Zros at,, - Pols at.75,.5 ±. 5 # A filtr is charactrid by followig ols ad ros o -la. Zros at Pols at Radius Agl Radius Agl. rad.5 rad rad Shows -la lot ad lot magitud rsos ot to scal. Solutio:- Zros ar, Pols ar: By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 57

58 Dowloadd from I m R Fig: -la lot H Put, H H H wt, wt wt wt wt..5 wt wt H WT WT /.8 WT /.9 WT. wt H wt H wt.78 / / / ωτ Fig: magitud rsos otch filtr: Dowloadd from -58

59 Dowloadd from It is a filtr that cotai o or mor d otchs or idally rfct ulls i its frqucy rsos charactristics. otch filtr ar usful i may alicatio whr scific frqucy comot must b illumiats for xaml, istrumtatio i rcordig systm rquirs that owr li frqucy of 5 H ad its harmoics b illumiatd. H wt ω ω ωτ ull To crat a ull i frqucy rsos of filtr at frqucy w w simly itroduc a air of comlx ± wo cougat ro o th uit circl., Systm fuctio is giv by, w w w H b Dat: 66/9/ Rsos to comlx xotial sigal: Frqucy rsos fuctio:- Th iut-outut rlatioshi for LTI systm is u t h x..i - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 59 To dvlo a frqucy domai charactristics of th systm. Lt us xcit th systm with comlx xotials. w x A < <. Whr, A is amlitud ad w is orbitrary frqucy cofid to th frqucy itrval [, ] ow, y h h w [ A ] w w A.. w w h. H Th fuctio Hw xists if th systm is BIBO stabl, that is h <.5

60 Dowloadd from Th rsos of th systm to comlx xotial is giv by, w y AH w.6 W ot that th rsos is also i th form of comlx xotial with th sam frqucy as th iut but altrd by multilicativ factor Hw. Th multilicativ factor is calld ig valu of th systm i this cas a comlx xotial of th form is a ig fuctio of LTI systm ad Hw valuatd at frqucy of iut sigal is corrsodig ig valu. # Dtrmi th o/ squc of th systm with imuls rsos h u Wh th iut is / comlx xotial squc x A u < <. Solutio:- H. H w w At w, 6.6 H 5 Ad thrfor th o/ is y A y 5 A , < < # Dtrmi th rsos of th systm with imuls rsos h u to th iut sigal x 5si cos < < Solutio:- H H w Th first trm i a iut sigal is a fixd sigal comot corrsodig to w. Thus, Dowloadd from -6

61 H Th scod trm i x has frqucy. Thus, H Dowloadd from Fially th third trm i x has a frqucy w. Thus H. Hc th rsos of th systm to x is, y si 6.6 cos, < < 5 # A LTI systm is dscribd by followig diffrc quatio y ay bx < a <. a Dtrmi th magitud ad has of frqucy rsos Hw of th systm. b Choos th aramtr b so that th maximum valu of H w ad has of Hw for a.9. c Dtrmi th o/ of th systm to sigal for x 5 si cos. Chatr:- 5 Dat: 66/9/5 Discrt filtr structur:- Cascad form structur:- H Π H Whr is itgr art of / H b b b o a a H x H x x H y y y - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 6

62 Dowloadd from Dowloadd from -6 y x b -a -a b x - - Paralll form structur: A c H H H H y x C # Dtrmi th cascad aralll raliatio for th systm dscribd by th systm fuctio 8 H Cascad form:-

63 Dowloadd from - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ H H H H H Th cascad rlatioshi is y x 7/8 -/ -/ -/ - -/ Paralll form:- 8 H 8 H 8 * ,,, *

64 Dowloadd from H h h hm- y Cascad form structur:- H H Π H bo b b,... x x H H H y x - - b b b y Lattic form structur: Fir filtr with systm fuctio H A m,... m m m m m A α A o m m m hm, hm α m,, m m dg r of olyomial. Dowloadd from -6

65 m y x α x m Dowloadd from x α m α m y For m. y x α x st ordr lattic filtr. Figur: Dirct form raliatio. f o f y x g o - g o - g f x g x f f g y x x g f g x x rflctio cofficit α f o f f y x g o - g o - g - g - g y x α x α x f f g x m x x α, α - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 65

66 Dowloadd from Covrsio of lattic cofficit to dirct form filtr cofficits:- Dirct form filtr cofficit { α } ca b obtaid form lattic cofficits } usig th rlatios. A B A Am B m m m A m m B m m B m Rvrs olyomial of. m,,... m m,,... m A m # A giv -stag lattic filtr with cofficit cofficit for dirct form structur. A A B α, α, Corrsodig to sigl stag lattic filtr. { i,,. Dtrmi FIR filtr β A A A β 8 α, α, α Corrsodig to d ordr lattic form. 8 β 8 A A β α, α, α, α 8 Covrsio of dirct form FIR filtr cofficit to lattic cofficit: Am mbm Am m m,... α α m m m m m Dat: 66/9/7 Dowloadd from -66

67 Dowloadd from - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 67 # Dtrmi lattic Cofficit Corrsodig to FIR filtr with systm fuctio. 8 5 A H Solutio: α, 8 5 B 8 B Z A A α K, 8 B B K A A α Lattic ad lattic laddr structur for IIR systm:- Lt us bgi with all-ol systm with systm fuctio. m a H m.. Diffrc quatio for this systm is m x y a y. Wh. - f g - x g f O y g o K K - x f f o f g o - g f g o - y f o f g o - y x y- y yy-

68 Dowloadd from x f - f O - f O y g g - - g g - Fig: Two stag lattic structur - g o f o x f f g - g f - g - f o f g - g f o g o - y f o g o f g - g o - - y- y- x m H C m a Cm A Laddr Co-fficit, v C m m,,,, M C m m m c Z vm / Bm x f m - f O - f O y g g - - g g - - g o v v v o # Obtai th lattic laddr structur. H C A Solutio: Lattic cofficit:- Dowloadd from -68

69 Dowloadd from A B A α A B α B B A B. 676 α.676 Laddr cofficit:- C v C C C v B C C vb v 5.6 C C vb.58 v C v v.9 v 5.6 v.58 Fig: Lattic laddr structur. Dat: 66/9/ Aalysis of ssitivity to quatiatio of filtr cofficit:- - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 69

70 Lt us cosidr a gral FIR filtr with systm, m Dowloadd from b H. a With quatid cofficit cofficit th systm fuctio. H m b a Whr quatid cofficit { b } ca b rlasd to th uquatid cofficits { a } & { b } by th rlatio, a a a,,, b b b,,,... m Whr a } rrsts th quatiatio rrors. { Th domiator of H may b xrssd to th form, D a Π. Whr, } ar th ols of H { Similarly w ca xrss th domiator of H as, D Π.5 Whr,,,... Ad is th rror or rturtatio rsultig form th quatiatio of filtr cofficits. W shall rlat th rturtatio rrors i th a }. Th rturtatio rror i δ i δa i ca b xrssd as,. 6 { δi δa δ / δa δd / δ i. 7 Th total rturtatio rror, i i Π l, l i i r q. 8 Th rror ca b miimid by maximiig th lgths th high ordr filtr with ithr sigl ol or doubl ol filtr sctios. i l. This ca b accomlishd by raliig Dowloadd from -7

71 Dowloadd from Limit cycl oscillatio i rcursiv systm:- I th raliatio of digital filtr ithr i digital hardwar or softwar o a digital comutr, th quatiatio ihrt i th fiit rcisio arithmtic oratio rdr th systm o liar. I rcursiv systm th o liaritis du to fiit rcisio arithmtic oratio oft caus riodic oscillatio to occur i o/ v wh th iut squc is ro or som o ro costat valu. Such oscillatio i rcursiv systm ar calld limit cycl ad ar dirctly attributabl to roud off rrors i multilicatio ad ovrflow rror i additio. Scalig to rvt ovrflow:- I ordr to limit th amout of o liar distortio it is imortat to scal th iut sigal ad uit saml rsos btw th iut ad ay itral summig mod i th systm such that ovrflow bcoms a rar vt. For fixd oit arithmtic lt us first cosidr th xtra coditio that ovrflow is ot rmittd at ay od of th systm. Lt y dot th rsos of th systm at th od wh iut squc is x ad uit saml rsos btw th od ad th iut i h. y h m x m h m x m m m Suos that x is ur boudd by A. Th, m y A h m for all ow, if th dyamic rag of th comutr is limitd to -, th coditio, y < Ca b satisfid by rquirig that th iut x ca b such that, Ax < h m m For all ossibl ods i th systm. Th coditio i is both cssary ad sufficit to rvt cofficit. For FIR filtr, bcom, A x < m..5 h m m Aothr aroach to scalig is to scal th iut so that, y C x C E.. 6 From arsavals thorm, y H w w E x H w dw.. 7 x - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 7

72 Dowloadd from Dowloadd from -7 From 6 ad 7 dw w H E E C x x dw w H C. 8 Rad th followig toics your slf:- Rrstatio of umbr. Fixd oit rrstatio. Floatig oit rrstatio. Roudig ad trucatio rror. Chatr: 6 Dsig of IIR filtr form aalog filtrs:- Imuls ivariac mthod: M i i i a s A s H Taig ivrs lalac trasform M i a t i a t u A t h i Put t T M i a T i a T u A T h i h H T u A M i a T i i M i a T i T u A i M i T i A i Hc, s T i i

73 Dowloadd from Q. Covrt th aalog filtr with th systm fuctio of imuls ivariac mthod. Solutio: s. s. H a s s. 9 s. s. * s. s., * H a s s. s. Th usig imuls ivariac mthod, H. T. T.T cost.t.t cost s. H a s ito a digital IIR by mas s. 9 IIR filtr dsig by biliar trasformatio: IIR filtr dig usig imuls ivariac ad aroximatio of drivativ mthod hav a svr limitatio i that thy ar aroximat oly for low ass filtr ad limitd calls of badass filtr. I this w dscrib a maig from s-la to -la calld biliar trasformatio that ovrcoms th limitatio of othr two dsig mthod dscrib rviously. Lt us cosidr th aalog filtr with systm fuctio. b H s.. s a Y s b s s a Or, sy s ay s b s Taig ivrs lalac trasform, w gt, du t ay t bx t dt Taig itgratio o both sids. du t dt a dt T mt y t dt Usig Traoidal rul, T g T T Th, T T T x t dt [ g T g T ] t dt T / T - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 7

74 Dowloadd from y T T at / [ y T y T ] bt / [ x T x T T ] y T T Taig -trasform o both sids, ` Y Y at / Y Y bt / [ ] [ ] / Y bt / at b H a T Hc, s T It givs th maig form s-la to -la. This is calld biliar trasformatio. To ivstigat th charactristics of biliar trasformatio lt, w r s σ Ω ow, s T w r w r r r si w T r r cos w Comarig with, s σ Ω σ T r Ω T r r r cos w r si w r cos w If r <, th σ < ad if r >, th, σ > cosqutly, th LHP is S mas ito isid a uit circl i -la ad RHP i s mas ito outsid a uit circl. Wh, r, σ, si w Ω T cos w Ω ta w T ΩT w ta Dowloadd from -7

75 Dowloadd from ω ω ta - ΩT/ / ΩT / Fig: maig btw frqucy variabls w ad Ω rsultig from biliar trasformatio. s. # Covrt th aalog filtr with systm H a s ito digital IIR filtr by mas of s. 6 biliar trasformatio th digital is to hav rsoat frqucy of w r /. Solutio: Th rsoat frqucy of aalog filtr is Ω r. & rsoat frqucy of digital filtr is Ω r ta w r / T ta T T ow, s / T. Usig biliar, th systm fuctio of digital filtr bcoms H H.95 - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 75

76 Dowloadd from Q. Dsig a sigl ol low ass digital filtr factor with db badwidth of. usig th biliar Ωc trasformatio aly to th aalog filtr H s. Whr, Ω c is a db badwidth aalog filtr. s Ωc Solutio: w c., For aalog filtr. Ω c ta w c / T ta. Ω T.65 T Aalog filtr has systm fuctio.65 T H s s.6 T ow, s T Th,.65 T H.65T T T, H.59 Matchd -trasformatio:- Aothr mthod for covrtig aalog filtr ito quivalt digital filtr is to ma th ols ad ros of Hs dirctly ito ols ad ro i -la. Suos th trasfr fuctio of aalog filtr is xrssd i th factord formd. M Π s H s Π Whr, { } s ar ros ad ols of th aalog filtr. Th systm fuctio for digital filtr is M T Π H s T Π ad { } Thus ach factor of th form s-a i Hs is mad ito th factor at. This maig is calld matchd -trasformatio. Dowloadd from -76

77 Dowloadd from # Covrt th aalog filtr with systm fuctio -trasformatio mthod. H a s. s ito digital IIR filtr by matchd s. 9 s. s. H a s s. s. s. Usig matchd -trasformatio mthod..t. T. T H.T.T T.T T Dat: 66//7 Buttr worth filtr: W hav, T w ε ω Wh ε T w ω This fuctio is ow as Buttrworth rsos. From this quatio w obsrv som itrstig rortis of Buttrworth rsos, ω. Th Buttrworth filtr is a all ol filtr. It has ro at ifiity T for all. T. 77 for all. corrsodig to db. For larg ω, T ω xhibits -ol roll-off. Tω ω Fig: Maximally flat rsos. - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 77

78 Dowloadd from Buttrwoth filtr:- α ω logt w db Low ass filtr scificatios, Passbad frqucy Ω Stobad frqucy Ω s Passbad attuatio α Stobad attuatio Ordr of filtr,.α s log. α log Ω Ω -db cutoff frqucy, Ω Ωc.α s α s Chbyshv filtr: W hav chbyshv magitud rsos, T w ε C ω C ω cos cos ω for ω cosh cosh ω for ω > With quatio, th trasfr fuctio magitud i is dtrmid for all valus of ω. Th fuctio is lottd for 6. odd bgi hr v bgi hr Fig: Sixth ordr chbyshv T F magitud C for odd ad C for v. T for odd. T for v. ε Dowloadd from -78

79 At ω, C T for all. ε.α s cosh.α cosh Ω Ω s Dowloadd from - qual ril filtr. - High attuatio i sto bad ad str roll-off ar th cut-off frqucy. * For Ω, Ωs. ad α.db α s db Which filtr is bst. Solutio, For buttrworth filtr 5. for chbyshv filtr. Gratr fficicy of chbyshv filtr comard with Buttrworth filtr. Frqucy trasformatio:- If w wish to dsig a high ass or badass or badsto filtr it is a siml mthod to ta a low ass rototy filtr buttrwoth, chbyshv rform a frqucy trasformatio. O ossibility is to rform th frqucy trasformatio i aalog domai ad th to covrt aalog filtr ito corrsodig digital filtr by a maig of s-la ito -la ad altrativ aroach is first to covrt th aalog low ass filtr ito digital low ass filtr ito a dsird digital filtr by a digital trasformatio. I gral, ths two aroachs yilds diffrt rsults xct for biliar trasformatio i which cas th rsultig filtr dsigs ar idtical. Frqucy trasformatio i aalog domai:- Ty of trasformatio Trasformatio Low ass Ω S s Ω High ass Bad ass Ω S S Ω ' Ω ' s s ΩLΩ Ω Ω s u L Bad sto s Ω Ω S Bad dg frq. of w filtr. Ω - By Er. Mao Bast Ass. Lcturr Eastr Collg of Egirig, Biratagar/ - 79 u ' Ω ' Ω L, Ω u Ω, Ω u L L u Ω s ΩLΩu Q. Trasform th sigl ol low ass buttrworth filtr with th systm fuctio bad ass filtr with ur ad lowr bad dg frqucy Solutio: Ωu ad H ΩL rsctivly. Ω s ito s Ω