EC1305 SIGNALS & SYSTEMS

Transcription

1 EC35 SIGNALS & SYSTES DEPT/ YEAR/ SE: IT/ III/ V PREPARED BY: s. S. TENOZI/ Lcturr/ECE

2 SYLLABUS UNIT I CLASSIFICATION OF SIGNALS AND SYSTES Cotiuous Tim Sigals (CT Sigals Discrt Tim Sigals (DT Sigals Stp Ramp Puls Impuls Expotial Classificatio of CT ad DT Sigals Priodic ad apriodic Radom Sigals CT systms ad DT systms Classificatio of systms Liar tim ivariat systms. UNIT II ANALYSIS OF CT SIGNALS Fourir sris aalysis Spctrum of CT sigals Fourir trasform ad laplac trasform i sigal aalysis. UNIT III LTI CT SYSTES Diffrtial quatio Bloc diagram rprstatio Impuls rspos Covolutio itgral Frqucy rspos Fourir mthods ad laplac trasforms i aalysis Stat quatios ad matrix. UNIT IV ANALYSIS OF DT SIGNALS Spctrum of DT sigals Discrt Tim Fourir Trasform (DTFT Discrt Fourir Trasform (DFT Proprtis of trasform i sigal aalysis. UNIT V LTI DT SYSTES Diffrc quatios Bloc diagram rprstatio Impuls rspos covolutio SU Frqucy rspos FFT ad - Trasform aalysis Stat variabl quatio ad matrix. TEXT BOOK. Ala V. Opphim, Ala S. Willsy ad S.amid Nawab, Sigals & Systms, Parso / Prtic all of Idia, 3. REFERENCES. K.Lidr, Sigals ad Systms, Tata cgraw-ill, Simo ayi ad Barry Va V, Sigals ad Systms, Joh Wily & Sos, 999.

3 UNIT I CLASSIFICATION OF SIGNALS AND SYSTES

4 SIGNAL: Sigal is a physical quatity that varis with rspct to tim, spac or ay othr idpdt variabl Eg x(t= si t. th maor classificatios of th sigal ar: (i Discrt tim sigal (ii Cotiuous tim sigal UNIT STEP &UNIT IPULSE Discrt tim Uit impuls is dfid as δ *+=,, {, = Uit impuls is also ow as uit sampl. Discrt tim uit stp sigal is dfid by U[]={,= {,>= Cotiuous tim uit impuls is dfid as δ (t=,, t=,, t Cotiuous tim Uit stp sigal is dfid as U(t={, t<,, t Classificatio of CT sigals.

5 Th CT sigals ar classifid as follows (i Priodic ad o priodic sigals (ii Ev ad odd sigals (iii Ergy ad powr sigals (iv Dtrmiistic ad radom sigals. Priodic Sigal & Apriodic Sigal A sigal is said to b priodic, if it xhibits priodicity. i.., X(t +T=x(t, for all valus of t. Priodic sigal has th proprty that it is uchagd by a tim shift of T. A sigal that dos ot satisfy th abov priodicity proprty is calld a apriodic sigal Ev ad odd sigal? A discrt tim sigal is said to b v wh, x[-]=x[]. Th cotiuous tim sigal is said to b v wh, x(-t= x(t For xampl, Cosω is a v sigal. ENERGY & POWER SIGNAL: A sigal is said to b rgy sigal if it hav fiit rgy ad ro powr. A sigal is said to b powr sigal if it hav ifiit rgy ad fiit powr. If th abov two coditios ar ot satisfid th th sigal is said to b ithr rgy or powr sigal SYSTE A systm is a st of lmts or fuctioal bloc that ar

6 coctd togthr ad producs a output i rspos to a iput sigal. Eg: A audio amplifir, attuator, TV st tc. Classificatio or charactristics of CT ad DT systms. Th DT ad CT systms ar accordig to thir charactristics as follows (i. Liar ad No-Liar systms (ii. Tim ivariat ad Tim varyig systm (iii. Causal ad No causal systms. (iv. Stabl ad ustabl systms. (v. Static ad dyamic systms. (vi. Ivrs systms. liar ad o-liar systms. A systm is said to b liar if suprpositio thorm applis to that systm. If it dos ot satisfy th suprpositio thorm, th it is said to b a o- liar systm. Causal ad o-causal systms. A systm is said to b a causal if its output at aytim dpds upo prst ad past iputs oly. A systm is said to b o-causal systm if its output dpds upo futur iputs also. Tim ivariat ad tim varyig systms. A systm is tim ivariat if th tim shift i th iput sigal rsults i corrspodig tim shift i th output. A systm which dos ot satisfy th abov coditio is tim variat systm. Stabl ad ustabl systms. Wh th systm producs boudd output for boudd iput, th th systm is calld boudd iput, boudd output stabl. A systm which dos ot satisfy th abov coditio is calld a ustabl systm. Static ad Dyamic systm.

7 A systm is said to b static or mmorylss if its output dpds upo th prst iput oly. Th systm is said to b dyamic with mmory if its output dpds upo th prst ad past iput valus.

8 IPORTANT QUESTIONS. Dfi Sigal.. Dfi Systm. 3. Dfi CT sigals. 4. Dfi DT sigal. 5. Giv fw xampls for CT sigals. 6. Giv fw xampls of DT sigals. 7. Dfi uit stp,ramp ad dlta fuctios for CT. 8. Stat th rlatio btw stp, ramp ad dlta fuctios(ct. 9. Stat th classificatio of CT sigals.. Dfi dtrmiistic ad radom sigals.. Dfi powr ad rgy sigals.. Compar powr ad rgy sigals. 3.Dfi odd ad v sigal. 4. Dfi priodic ad Apriodic sigals. 5. Stat th classificatio or charactristics of CT ad DT systms. 6. Dfi liar ad o-liar systms. 7. Dfi Causal ad o-causal systms. 8. Dfi tim ivariat ad tim varyig systms. 9. Dfi stabl ad ustabl systms.. Dfi Static ad Dyamic systm. PART-B. Discuss th classificatio of DT ad CT sigals with xampls.. Discuss th classificatio of DT ad CT systms with xampls. 3. Problms o th proprtis & classificatios of sigals & systms Fid whthr th followig sigals ar priodic or ot a. x(t=cos(t+-si(4t- As:Priodic sigal. b. x(t=3cos4t+siπt As:No priodic sigal Chc whthr th followig systm is. Static or dyamic. Liar or o-liar 3. Causal or o-causal 4. Tim ivariat or variat y(=sg[x(]

9 UNIT II ANALYSIS OF CT SIGNALS

10 FOURIER SERIES: Th Fourir sris rprsts a priodic sigal i trms of frqucy compots: x( p i X x t X i t ( W gt th Fourir sris cofficits as follows: X p p x( i X p p x( t i t dt Th complx xpotial Fourir cofficits ar a squc of complx umbrs rprstig th frqucy compot ω. Fourir sris: a complicatd wavform aalyd ito a umbr of harmoically rlatd si ad cosi fuctios A cotiuous priodic sigal x(t with a priod T may b rprstd by: X(t=Σ = (A cos ω t + B si ω t+ A Dirichlt coditios must b placd o x(t for th sris to b valid: th itgral of th magitud of x(t ovr a complt priod must b fiit, ad th sigal ca oly hav a fiit umbr of discotiuitis i ay fiit itrval TRIGONOETRIC FOR OF FOURIER SERIES If th two fudamtal compots of a priodic sigal arbcosωt ad Csiωt, th thir sum is xprssd by trigoomtric idtitis:

11 X(t= A + Σ = ( B + A / (C cos ω t- φ or X(t= A + Σ = ( B + A / (C si ω t+ φ FOURIER TRANSFOR: Viwd priodic fuctios i trms of frqucy compots (Fourir sris as wll as ordiary fuctios of tim Viwd LTI systms i trms of what thy do to frqucy compots (frqucy rspos Viwd LTI systms i trms of what thy do to tim-domai sigals (covolutio with impuls rspos Viw apriodic fuctios i trms of frqucy compots via Fourir trasform Dfi (cotiuous-tim Fourir trasform ad DTFT Gai isight ito th maig of Fourir trasform through compariso with Fourir sris A trasform tas o fuctio (or sigal ad turs it ito aothr fuctio (or sigal Cotiuous Fourir Trasform: f h t ift dt h t CONTINUOUS TIE FOURIER TRANSFOR W ca xtd th formula for cotiuous-tim Fourir sris cofficits for a priodic sigal to apriodic sigals as wll. Th cotiuous-tim Fourir sris is ot dfid for apriodic f ift df

12 sigals, but w call th formula th (cotiuous tim Fourir trasform. INVERSE TRANSFORS If w hav th full squc of Fourir cofficits for a priodic sigal, w ca rcostruct it by multiplyig th complx siusoids of frqucy ω by th wights X ad summig: W ca prform a similar rcostructio for apriodic sigals Ths ar calld th ivrs trasforms. FOURIER TRANSFOR OF IPULSE FUNCTIONS: Fid th Fourir trasform of th Dirac dlta fuctio: Fid th DTFT of th Krocr dlta fuctio: / / ( ( p p t i p t i dt t x p dt t x p X dt t x X t i ( ( ( p i X x t i X t x ( d X x i ( ( d X t x t i ( ( ( ( ( i t i t i dt t dt t x X ( ( ( i i i x X

13 Th dlta fuctios cotai all frqucis at qual amplituds. Roughly spaig, that s why th systm rspos to a impuls iput is importat: it tsts th systm at all frqucis. LAPLACE TRANSFOR Lapalc trasform is a graliatio of th Fourir trasform i th ss that it allows complx frqucy whras Fourir aalysis ca oly hadl ral frqucy. Li Fourir trasform, Lapalc trasform allows us to aaly a liar circuit problm, o mattr how complicatd th circuit is, i th frqucy domai i stad of i h tim domai. athmatically, it producs th bfit of covrtig a st of diffrtial quatios ito a corrspodig st of algbraic quatios, which ar much asir to solv. Physically, it producs mor isight of th circuit ad allows us to ow th badwidth, phas, ad trasfr charactristics importat for circuit aalysis ad dsig. ost importatly, Laplac trasform lifts th limit of Fourir aalysis to allow us to fid both th stady-stat ad trasit rsposs of a liar circuit. Usig Fourir trasform, o ca oly dal with h stady stat bhavior (i.. circuit rspos udr idfiit siusoidal xcitatio. Usig Laplac trasform, o ca fid th rspos udr ay typs of xcitatio (.g. switchig o ad off at ay giv tim(s, siusoidal, impuls, squar wav xcitatios, tc.

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16 APPLICATION OF LAPLACE TRANSFOR TO CIRCUIT ANALYSIS

17 CONVOLUTION Th iput ad output sigals for LTI systms hav spcial rlatioship i trms of covolutio sum ad itgrals. Y(t=x(t*h(t Y[]=x[]*h[]

18 IPORTANT QUESTIONS.Dfi CT sigal. Compar doubl sidd ad sigl sidd spctrums. 3. Dfi Quadratur Fourir Sris. 4.Dfi polar Fourir Sris. 5.Dfi xpotial fourir sris. 6. Stat Dirichlts coditios. 7. Stat Parsvals powr thorm. 8.Dfi Fourir Trasform. 9. Stat th coditios for th xistc of fourir sris.. Fid th Fourir trasform of fuctio x(t=δ(t. Stat Rayligh s rgy thorm..dfi laplac trasform. 3. Obtai th laplac trasform of ramp fuctio. 4. What ar th mthods for valuatig ivrs Laplac trasform. 5. Stat iitial valu thorm. 6. Stat fial valu thorm. 7. Stat th covolutio proprty of fourir trasform. 8.What is th rlatioship btw Fourir trasform ad Laplac trasform. 9.Fid th fourir trasform of sg fuctio.. Fid out th laplac trasform of f(t= at PART- B.Stat ad prov proprtis of fourir trasform.. Stat th proprtis of Fourir Sris. 3. Stat th proprtis of Laplac trasform. 4.Problms o fourir sris, Fourir trasform ad laplac trasform. a. Fid th fourir sris of of th priodic sigal x(t=t <=t<= b. Fid th fourir trasform of x(t= -at u(t c. Fid th laplac trasform of th sigal x(t= -at u(t+ -bt u(-t 5. Stat ad prov parsvals powr thorm ad Rayligh s rgy thorm.

19 UNIT III LTI-CT SYSTES

20 SAPLING TEORY Th thory of taig discrt sampl valus (grid of color pixls from fuctios dfid ovr cotiuous domais (icidt radiac dfid ovr th film pla ad th usig thos sampls to rcostruct w fuctios that ar similar to th origial (rcostructio. Samplr: slcts sampl poits o th imag pla Filtr: blds multipl sampls togthr For bad limitd fuctio, w ca ust icras th samplig rat owvr, fw of itrstig fuctios i computr graphics ar bad limitd, i particular, fuctios with discotiuitis.

21 It is bcaus th discotiuity always falls btw two sampls ad th sampls provids o iformatio of th discotiuity.

22 ALIASING

23 Z-TRANSFORS For discrt-tim systms, -trasforms play th sam rol of Laplac trasforms do i cotiuous-tim systms Bilatral Forward -trasform Bilatral Ivrs -trasform [ ] h R h[ ] [ ] d As with th Laplac trasform, w comput forward ad ivrs -trasforms by us of trasforms pairs ad proprtis

24 REGION OF CONVERGENCE Rgio of th complx -pla for which forward -trasform covrgs Four possibilitis (= is a spcial cas ad may or may ot b icludd

25 Z-TRANSFOR PAIRS h[] = d[] [ ] Rgio of covrgc: tir -pla h[] = d[-] [ ] Rgio of covrgc: tir -pla h[-] - [] h[] = a u[] Rgio of covrgc: > a which is th complmt of a dis STABILITY Rul #: For a causal squc, pols ar isid th uit circl (applis to -trasform fuctios that ar ratios of two polyomials Rul #: or grally, uit circl is icludd i rgio of covrgc. (I cotiuous-tim, th imagiary axis would b i th rgio of covrgc of th Laplac trasform.

26 This is stabl if a < by rul #. It is stabl if > a ad a < by rul #. INVERSE Z-TRANSFOR f c c F d Yu! Usig th dfiitio rquirs a cotour itgratio i th complx -pla. Fortuatly, w td to b itrstd i oly a fw basic sigals (puls, stp, tc. Virtually all of th sigals w ll s ca b built up from ths basic sigals. For ths commo sigals, th -trasform pairs hav b tabulatd (s Lathi, Tabl 5.

27 EXAPLE Ratio of polyomial -domai fuctios Divid through by th highst powr of Factor domiator ito firstordr factors Us partial fractio dcompositio to gt first-ordr trms Fid B by polyomial divisio Exprss i trms of B Solv for A ad A 3 ] [ X 3 ] [ X ] [ X ] [ A A B X ] [ X A A

28 Exprss X[] i trms of B, A, ad A Us tabl to obtai ivrs -trasform With th uilatral -trasform, or th bilatral -trasform with rgio of covrgc, th ivrs -trasform is uiqu 8 9 ] [ X u u x 8 9

29 Covolutio dfiitio Ta -trasform Z-trasform dfiitio Itrchag summatio Substitut r = - m Z-trasform dfiitio F F r f m f r f m f m f m f m f m f m f m f Z f f Z m f m f f f r r m m m r m r m m m m

30 IPORTANT QUESTIONS. Dfi LTI-CT systms.. What ar th tools usd for aalysis of LTI-CT systms? 3.Dfi covolutio itgral. 4.List th proprtis of covolutio itgral. 5.Stat commutativ proprty of covolutio. 6.Stat th associativ proprty of covolutio. 7.Stat distributiv proprty of covolutio. 8. Wh th LTI-CT systm is said to b dyamic? 9. Wh th LTI-CT systm is said to b causal?. Wh th LTI-CT systm is said to b stabl?. Dfi atural rspos.. Dfi forcd rspos. 3. Dfi complt rspos. 4. Draw th dirct form I implmtatio of CT systms. 5. Draw th dirct form II implmtatio of CT systms. 6. tio th advatags of dirct form II structur ovr dirct form I structur. 7. Dfi Eig fuctio ad Eig valu. 8. Dfi Causality ad stability usig pols. 9. Fid th impuls rspos of th systm y(t=x(t-t usig laplac trasform.. Th impuls rspos of th LTI CT systm is giv as h(t= -t u(t. Dtrmi trasfr fuctio ad chc whthr th systm is causal ad stabl. PART B.Driv covolutio itgral ad also stat ad prov th proprtis of th sam.. Explai th proprtis of LTICT systm itrms of impuls rspos. 3.Problms o proprtis of LTI CT systms. 4. Problms o diffrtial quatio. 5. Raliatio of LTI CT systm usig dirct form I ad II structurs. 6. Fidig frqucy rspos usig Fourir mthods. 7. Solvig diffrtial quatios usig Fourir mthods 8. Solvig Diffrtial Equatios usig Laplac trasforms. 9. Obtaiig stat variabl dscriptio.. Obtaiig frqucy rspos ad trasfr fuctios usig stat variabl.

31 UNIT IV ANALYSIS OF DISCRETE TIE SIGNALS

32 INTRODUCTION Impuls rspos h[] ca fully charactri a LTI systm, ad w ca hav th output of LTI systm as Th -trasform of impuls rspos is calld trasfr or systm fuctio (. Frqucy rspos at is valid if ROC icluds ad FREQUENCY RESPONSE OF LIT SYSTE Cosidr ad, th magitud phas W will modl ad aaly LTI systms basd o th magitud ad phas rsposs. SYSTE FUNCTION Gral form of LCCDE ( ( ( X X X ( ( ( X Y ( ( ( X Y h x y. X Y, X Y ( ( ( x b y a N X b Y a N (

33 Comput th -trasform SYSTE FUNCTION: POLE/ZERO FACTORIZATION Stability rquirmt ca b vrifid. Choic of ROC dtrmis causality. Locatio of ros ad pols dtrmis th frqucy rspos ad phas SECOND-ORDER SYSTE Suppos th systm fuctio of a LTI systm is N a b X Y N d c a b.,...,, : ros c c c.,...,, : pols N d d d. 4 3 ( ( ( (

34 To fid th diffrc quatio that is satisfid by th iput ad out of this systm ( ( ( ( Y ( X ( 3 y[ ] y[ ] y[ ] x[ ] x[ ] x[ 4 8 ] Ca w ow th impuls rspos? Systm Fuctio: Stability Stability of LTI systm: h [ ] This coditio is idtical to th coditio that h[ ] wh. Th stability coditio is quivalt to th coditio that th ROC of ( icluds th uit circl. Systm Fuctio: Causality If th systm is causal, it follows that h[] must b a right-sidd squc. Th ROC of ( must b outsid th outrmost pol. If th systm is ati-causal, it follows that h[] must b a lft-sidd squc. Th ROC of ( must b isid th irmost pol.

35 DETERINING TE ROC Cosidr th LTI systm 5 y[ ] y[ ] y[ ] x[ ] Th systm fuctio is obtaid as ( ( 5 ( SYSTE FUNCTION: INVERSE SYSTES i is a ivrs systm for, if G( ( i( g h h i ( i i( ( ( Th ROCs of ( ad ( i must ovrlap. Usful for caclig th ffcts of aothr systm S th discussio i Sc.5.. rgardig ROC

36 ALL-PASS SYSTE A systm of th form (or cascad of ths Ap Z a a pol : ro : a r / a* r Ap a a a * a Ap ALL-PASS SYSTE: GENERAL FOR I gral, all pass systms hav form Ap d d r c ( ( * ( ( * ral pols complx pols Causal/stabl:, d

37 INIU-PASE SYSTE iimum-phas systm: all ros ad all pols ar isid th uit circl. Th am miimum-phas coms from a proprty of th phas rspos (miimum phas-lag/group-dlay. iimum-phas systms hav som spcial proprtis. Wh w dsig a filtr, w may hav multipl choics to satisfy th crtai rquirmts. Usually, w prfr th miimum phas which is uiqu. All systms ca b rprstd as a miimum-phas systm ad a all-pass systm.. Dfi DTFT.. Stat th coditio for xistc of DTFT? 3. List th proprtis of DTFT. 4. What is th DTFT of uit sampl? 5. Dfi DFT. 6. Dfi Twiddl factor. 7. Dfi Zro paddig. 8. Dfi circularly v squc. 9. Dfi circularly odd squc.. Dfi circularly foldd squcs.. Stat circular covolutio.. Stat parsval s thorm. 3. Dfi Z trasform. 4. Dfi ROC. 5. Fid Z trasform of x(={,,3,4} 6. Stat th covolutio proprty of Z trasform. 7. What trasform of (-m? 8. Stat iitial valu thorm. 9. List th mthods of obtaiig ivrs Z trasform. IPORTANT QUESTIONS PART-A

38 . Obtai th ivrs trasform of X(=/- PART B. Stat ad prov proprtis of DTFT. Stat ad prov th proprtis of DFT. 3. Stat ad prov th proprtis of trasform. 4.Fid th DFT of x(={,,,,,,,} 5. Fid th circular covolutio of x (={,,,} X (={,,,} 6. Problms o trasform ad ivrs trasform.

39 UNIT V LTI DT SYSTE

40 Exampl y a y a y b x Bloc diagram rprstatio of BLOCK DIAGRA REPRESENTATION LTI systms with ratioal systm fuctio ca b rprstd as costatcofficit diffrc quatio Th implmtatio of diffrc quatios rquirs dlayd valus of th iput output itrmdiat rsults Th rquirmt of dlayd lmts implis d for storag W also d mas of additio multiplicatio

41 DIRECT FOR I Gral form of diffrc quatio Altrativ quivalt form ALTERNATIVE REPRESENTATION N x b y a ˆ ˆ N x b y a y

42 Rplac ordr of cascad LTI systms ALTERNATIVE BLOCK DIAGRA W ca chag th ordr of th cascad systms W b W X a X a b N N N w b y x w a w N w b y x w a w

43 DIRECT FOR II No d to stor th sam data twic i prvious systm So w ca collaps th dlay lmts ito o chai This is calld Dirct Form II or th Caoical Form Thortically o diffrc btw Dirct Form I ad II Implmtatio wis Lss mmory i Dirct II Diffrc wh usig fiit-prcisio arithmtic SIGNAL FLOW GRAP REPRESENTATION Similar to bloc diagram rprstatio Notatioal diffrcs A twor of dirctd brachs coctd at ods Exampl rprstatio of a diffrc quatio

44 EXAPLE Rprstatio of Dirct Form II with sigal flow graphs w y w w b w w b w w w x aw w b w w b y x aw w

45 DETERINATION OF SYSTE FUNCTION FRO FLOW GRAP w w y w w x w w w w x w w W W W X W W X W W W Y X W X W 4 4 u u h X Y

46 BASIC STRUCTURES FOR IIR SYSTES: DIRECT FOR I BASIC STRUCTUR ES FOR IIR SYSTES: DIRECT FOR II

47 BASIC STRUCTURES FOR IIR SYSTES: CASCADE FOR Gral form for cascad implmtatio or practical form i d ordr systms EXAPLE N N d d c g g f A a a b b b

48 CASCADE OF DIRECT FOR I SUBSECTIONS Cascad of Dirct Form II subsctios BASIC STRUCTURES FOR IIR SYSTES: PARALLEL FOR REPRESENT SYSTE FUNCTION USING PARTIAL FRACTION EXPANSION Or by pairig th ral pols P N P P N N d d B c A C P N S N a a C

49 EXAPLE Partial Fractio Expasio Combi pols to gt BASIC STRUCTURES FOR FIR SYSTES: DIRECT FOR

50 Spcial cass of IIR dirct form structurs Traspos of dirct form I givs dirct form II Both forms ar qual for FIR systms Tappd dlay li BASIC STRUCTURES FOR FIR SYSTES: CASCADE FOR Obtaid by factorig th polyomial systm fuctio h S b b b STRUCTURES FOR LINEAR-PASE FIR SYSTES Causal FIR systm with gralid liar phas ar symmtric: h h,,..., (typ I or III h h,,..., (typ II or IV

51 Symmtry mas w ca half th umbr of multiplicatios Exampl: For v ad typ I or typ III systms: STRUCTURES FOR LINEAR-PASE FIR SYSTES Structur for v Structur for odd / / / / / / / / / / / x h x x h x h x h x h x h x h x h x h y

52 IPORTANT QUESTIONS PART-A. Dfi covolutio sum?. List th stps ivolvd i fidig covolutio sum? 3. List th proprtis of covolutio? 4. Dfi LTI causal systm? 5. Dfi LTI stabl systm? 6. Dfi FIR systm? 7. Dfi IIR systm? 8. Dfi o rcursiv ad rcursiv systms? 9. Stat th rlatio btw fourir trasform ad trasform?. Dfi systm fuctio?. What is th advatag of dirct form ovr dirct form structur?. Dfi buttrfly computatio? 3.What is a advatag of FFT ovr DFT? 4. List th applicatios of FFT? 5. ow uit sampl rspos of discrt tim systm is dfid? 6.A causal DT systm is BIBO stabl oly if its trasfr fuctio has. 7. If u( is th impuls rspos of th systm, What is its stp rspos? 8.Covolv th two squcs x(={,,3} ad h(={5,4,6,} 9. Stat th maximum mmory rquirmt of N poit DFT icludig twiddl factors?. Dtrmi th rag of valus of th paramtr a for which th liar tim ivariat systm with impuls rspos h(=a u( is stabl? PART-B. Stat ad prov th proprtis of covolutio sum?. Dtrmi th covolutio of x(={,,} h(=u( graphically? 3. Dtrmi th forcd rspos for th followig systm 4. Comput th rspos of th systm 5. Driv th 8 poit DIT ad DIF algorithms

53 UNIVERSITY QUESTIONS

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