SIGNALS AND LINEAR SYSTEMS UNIT-1 SIGNALS

Transcription

1 SIGNALS AND LINEAR SYSTEMS UNIT- SIGNALS. Dfi a sigal. A sigal is a fuctio of o or mor idpdt variabls which cotais som iformatio. Eg: Radio sigal, TV sigal, Tlpho sigal, tc.. Dfi systm. A systm is a st of lmts or fuctioal block that is coctd togthr ad producs a output i rspos to a iput sigal. Eg: Audio amplifir, attuator, TV, tc. 3. Dfi CT sigals. Cotiuous Tim Sigals (CT) ar dfid for all valus of tim. Its also calld as a aalog sigal ad is rprstd by x(t). Eg: AC wavform, ECG, EEG, tc. 4. Dfi DT sigals. Discrt Tim Sigals (DT) ar dfid at discrt istacs of tim. Its rprstd by x(). Eg: Amout dpositd i a bak pr moth. 5. Giv fw xampls for CT sigals. AC wavform, ECG, Tmpratur rcordd ovr a itrval. 6. Giv xampls for DT sigals. Amout dpositd i bak pr moth, Raifall rcordd i a city. 7. Dfi Uit Stp, Uit Ramp ad Uit Dlta fuctios for CT sigals.

2 8. Stat th classificatio of CT sigals. Th CT sigals ar classifid as follows: (i) Priodic ad o-priodic (apriodic) sigals. (ii) Ev ad odd sigals. (iii)ergy ad powr sigals. (iv) Dtrmiistic ad radom sigals. 9. Dfi dtrmiistic ad radom sigal. A dtrmiistic sigal is o which ca b compltly rprstd by mathmatical quatio at aytim. I a dtrmiistic sigal, thr is o ucrtaity with rspct to this valu at aytim. Eg: x(t)=cosωt,x()=cosω A radom sigal is o which caot b rprstd by a mathmatical quatio. Eg: Nois gratd i lctroic compots. 0. Dfi powr ad rgy sigals. Th sigal x(t) is said to b powr sigal, if ad oly if th ormalid avrag powr is fiit ad oro. i.,0<p< Th sigal x(t) is said to b rgy sigal if ad oly ormalid total rgy is fiit ad oro.i,0<e<. Compar powr ad rgy sigals. S.No Ergy Sigal Powr Sigal. Ergy of th sigal is fiit ad oro. Powr of th sigal is fiit ad o-ro.. Powr of th sigal is ro. Ergy of th sigal is ifiit. 3. No-priodic sigals. Priodic sigals. Dfi odd ad v sigals. A DT sigal x() is said to b a v sigal if x()=x(-) ad a odd sigal if x()=-x(-)a CT sigal x(t) is said to b v sigal if x(t)=x(-t) ad odd sigal if x(t)=-x(-t).

3 3. Dfi priodic ad apriodic sigal (o-priodic) sigals. A sigal is said to b priodic sigal if it rpats at qual itrvals. Apriodic sigals do ot rpat at rgular itrvals. A CT sigal which satisfis th quatio x(t)=x(t+t) is said to b priodic. A DT sigal which satisfis th quatio x()=x(+n) is said to b priodic. 4. Stat th classificatio of CT ad DT systms. Th DT ad CT systms ar classifid accordig to thir charactristic as follows. (i) Liar ad o-liar systms. (ii) Causal ad o-causal systms. (iii)tim ivariat ad tim variat systms. (iv) Stabl ad ustabl systms. (v) Static ad dyamic systms. (vi) Ivrs systms. 5. Dfi liar ad No-Liar systms. A Systm is said to b Liar, if it satisfis th suprpositio thorm. Othrwis th Systm is said to b No-Liar systms. 6. Dfi Casual ad No-Casual systms. A Systm is said to b Casual, if its output at ay tim dpds o prst ad post iputs oly, A Systm is said to b No-Casual systm, if its output dpds o futur iputs also. 7. Dfi Tim Ivariat ad Tim Variat 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. 8. Dfi Stabl ad Ustabl systms. Wh th systm producs boudd output for vry boudd iput, th th systm is calld boudd iput boudd output (BIBO) stabl. A systm which dos ot satisfy th abov coditio is calld ustabl systm.

4 9. Dfi Static ad Dyamic systms. A systm is said to b Static or mmory lss, if its output dpds upo th prst iput oly. A Systm is said to b Dyamic systm, if its output dpds upo th prst ad past iput valus. 0. Cosidr th complx valud xpotial sigal x(t)= A whr α>0. Evaluat th Ral ad Imagiary Compots of x(t). Giv x(t)= A. x(t)= A. x(t)=a Ral part of x(t) is A (cos Ωt+ si Ωt). cos Ωt. Imagiary part of x(t) is A si Ωt.. Dtrmi whthr th sigal x(t)=3cos t+7si5πt is priodic. Giv that x(t)=3cos t+7si5πt Lt (t) = 3cos t ad (t)=7si5πt. Lt b priod of (t). O comparig (t) with stadard form Acosπft, w gt πf= => f= /π => = /f= π/. Lt b priod of (t). O comparig (t) with stadard form Asiπft, w gt Now, Hr, is ot a ratioal umbr ad so x(t) is a o-priodic sigal.

5 . Dtrmi th priod of th sigal Lt.O comparig x (t) with stadard form,w gt Lt comparig x t(t) with stadard form A si,w gt Hr, is a ratioal umbr. So x(t) is priodic sigal. Lt T b priod of x(t), which is giv by LCM of T ad T, LCM (3, ) = 6. So T = 6 3. Dtrmi th rgy ad powr of a uit stp sigal. Uit stp sigal Ergy, = = Powr, P=

6 P= ½ watts 4. Dtrmi th v ad odd part of complx xpot sigal. Complx xpotial sigal From this, Ev part of th sigal (t) = ½ A cos = A cos. Odd part of th sigal (t) = / - Thrfor (t) = / t + Asi t A cos t + Asi =/ = A si t 5. Sktch v ad odd parts of a uit stp sigal. Ev part v ( t) u( t) u( t)

7 Odd part v 0 ( t) u( t) u( t) 6. Diffrtiat btw causal ad o-causal sigals. Th causal sigals ar dfi oly for t 0 whras th o-causal sigals ar dfi for ithr t 0 or for all t.(i. both t 0 ad t 0). 7. A cotiuous tim sigal is show i fig. Fid th followig vrsio of th sigal. (a) x(-t) (b)- x(t).

8 8. A cotiuous tim sigal is show i fig. Fid th followig vrsio of th sigal. (a) x(t+) (b) x(-t) (c) x(-t-). 9. Evaluat th followig itgrals (i) ( t 3) t dt (ii) [ ( t )cost ( t )si t] dt (i) t t ( t 3) dt = ( t ( 3) dt = 3

9 = (ii) [ ( t )cost ( t )si t] dt =cost t=0 +sit t= =cos0 + si = If two LTI systms with impuls rsposs,h(t)= -at u(t) ad h(t)= -bt u(t) ar coctd i cascad, what will b th ovrall impuls rspos of cascadd systm? Wh two LTI systms ar i cascad, th ovrall impuls rspos is giv by covolutio of idividual rsposs. h 0 (t)=h (t) * h (t)= h (τ) h (t- τ)dτ h 0 (t)= h 0 (t)= t 0 -aτ -b(t-τ) dτ -aτ -bt bτ dτ = -bt -(a-b)τ dτ = -bt [ - (a-b) t / (a-b)] 0 = / (a-b) [- -at + -bt ] u[t] 3. What is th importac of covolutio? Th covolutio opratio ca b usd to dtrmi th rspos of a LTI systm for ay iput from th kowldg of its impuls rspos. Th rspos (output) of a LTI systm is giv by th covolutio of iput ad impuls rspos of LTI systm. 3. Chck whthr th giv systm is causal ad stabl y()=3x(-)+3x(+) Sic y() dpds upo x(+),this systm is o-causal.

10 As log as x(-) ad x(+) ar boudd, th output y() will b boudd. Hc th systm is stabl. PART B. Fid whthr th followig sigals ar priodic or ot. (a) x(t)=si 50 πt (b) x(t)=cos 60πt+si 50πt (c) x()=cos(π) (d) x()=cos(π/)-si(π/8)+3cos(π/4+π/3) Hit: (a) vrify whthr x(t)=x(t+t).if it is satisfid th sigal is priodic (b) fid T,T. Chck whthr T/T is ratioal. If it is ratioal th th sigal is priodic. (c) & (d) sam procdur as (a)&(b).. Discuss th classificatio of DT ad CT sigals with xampls: Hit: xplai th followig sigals: i) Dtrmiistic ad radom sigals. ii) Priodic ad apriodic sigals. iii) Ergy ad powr sigals. iv) Ev ad odd sigals v) Casual ad o-causal. 3. Discuss th classificatio of DT ad CT systms with xampls. Hit: xplai th followig systms with xampl. a. Liar ad o-liar systms b. Tim ivariat ad tim ivariat systms c. Causal ad o-causal systms

11 d. Static ad dyamic systms.. Stabl ad ustabl systms 4. Fid which of th followig sigals rgy ar or powr sigals. (a)x(t)= -3t u(t) (b)x(t)=cost (c)x()={,,} (d)x(t)= -πft Hit: E= T T x( t) dt ad P = T T T x( t) dt Usig this formula, fid rgy ad powr. If If 0<E<, th th sigal is rgy sigal. 0<P<, th th sigal is powr sigal. 5. Fid v ad odd compots of th followig sigal: (a) x(t)=cost+sit+cost.sit (b)x()={-,,, -,3} Hit: x (t)=/ {x(t)+x(-t)} x o (t)=/{x(t)-x(-t)} Usig this formula, Fid v ad odd part of th sigal. 6. For th sigal x(t) show, Fid th sigal x(t-),x(t+3),x(3t/),x(-t+). Hit: Whil doig tim trasformatio, it is importat to ot th ordr of prfrc as follows Tim rvrsal, Tim shiftig, Tim scalig.

12 7. Chck whthr th followig systms ar liar, tim ivariat, causal ad stabl. (i) y() = x() (ii) y(t) = (iii)y(+) = ax(+)+bx(+3) Hit: Us dfiitio to fid whthr th systm is liar, ivariat, causal ad stabl. 8. Fid th covolutio of x(t) ad h(t). x(t)= h(t)= Hit: (i) Fid x(τ) ad h(-τ). (ii) Fid h(t- τ) by shiftig h(-τ) t tims. (iii)multiply x(τ) ad h(t- τ). (iv) Itgrat th rsult. As: y(t) = x(t)*h(t) = 9. Fid th rspos, y(t) of a LTI systm whos x(t) ad h(t) ar show. As: y(t) = x(t)*h(t) = x ( ) h( t ) d

13 y(t) = 0. Dtrmi th complt rspos of th systm dy ( t) t=0 = ad x(t) = -t u(t). dt d y( t) dy( t) dx( t) 5 4y( t) with y(0) =0 dt dt dt As: Fid particular solutio,y p (t) = -t ; t 0. Fid homogous solutio, y =c -t +c -4t. Complt (Total) rspos y(t) = y p (t) v+ y (t) Usig iitial coditios, dtrmi c & c. y(t) = { - 4t -t -t } u(t).

14 UNIT- FOURIER ANALYSIS OF THE SIGNALS AND SYSTEMS PART A. Show that th vctors (,) ad (-3,) i R ar liarly idpdt? Th matrix formd by th vctor is A 3 W writ liar combiatio of th colum as AA= 3 Dt of A= 3 = -(-3)=5 0 Sic th dtrmiat is o-ro, th vctors (,) ad (-3,) ar liarly idpdt.. Dfi Bass? Lt V b a vctor spac ad S={V,V, V K } b a subst of V,th S is a basis for V if th followig two statmts ar tru, (i) S Spas V (ii) S is a liarly idpdt of vctors i V. 3. Dfi Dimsios? If S={V, V V } is a basis for a vctor spac V ad T={W,W W K } is a liarly idpdt vctors i V, th k. Th dimsios of a vctor V is th umbr of vctors i basis. 4. What is mat by orthogoal st? Two vctors ar said to b orthogoal if thir ir product (dot product) is ro.

15 5. Dfi mtric? A mtric o a st X is a fuctio d:x.x- R calld as distac fuctio or simply distac. 6. What is th coditio for xistc of Fourir sris? (i) Th fuctio x(t) should b sigl valud i ay fiit tim itrval T. (ii) Th fuctio x(t) should hav fiit umbr of discotiuitis i ay fiit tim itrval. (iii)th fuctio x(t) should hav fiit umbr of maxima ad miima i ay tim itrval T. (iv) Th fuctio x(t) should b absolutly itgrabl. 7. Compar doubl sidd ad sigl sidd spctrum. Th mthod of rprstig spctrums of positiv as wll as gativ frqucis ar calld doubl sidd spctrum. Th mthod of rprstig spctrums oly i th positiv frqucis is kow as sigl sidd spctrum. 8. Dfi quadratur Fourir sris. Cosidr x(t) b a priodic sigal. Th Fourir sris ca b writt as x(t)= + [a cos( 0 t)+b si( 0 t)] Whr =/T a =/T cos( 0t) dt b =/T si( 0t)dt. 9. Dfi xpotial Fourir sris. Cosidr x(t) b a priodic sigal. Th Fourir sris ca b writt as -i 0t x(t)= Whr C =i/t - 0t dt.

16 0. What is magitud ad phas spctrum? Magitud spctrum is th pol btw Magitud Vs Frqucy. Magitud= RP IP Phas spctrum is th plot btw phas aglvs Frqucy. Phas=. Stat Parsval s Thorm. Parsval s powr thorm stats that th total avrag of a priodic sigal x(t) is qual to th sum of avrag powr of its phasor compots.. Dfi Fourir Trasform. Lt x(t) b th sigal which is fuctio of tim t. Th Fourir trasform of x(t) is giv by X ( ) x( t) t dt 3. Dfi ivrs Fourir Trasform. Lt x( ) b th sigal which is fuctio of frqucy.th ivrs Fourir Trasform is giv by x( t) X ( ) d for all t. 4. Fid th Fourir trasform of fuctio x(t)= (t). Fourir trasform of a sigal x(t) is giv by, X( ) = x (t) - t dt. Substitutig x(t) = (t), X( ) = (t) - t dt. By usig proprty of impuls fuctio, (t) x(t) dt= x(0) X( ) =

17 5. Stat Rayligh`s rgy thorm. Rayligh`s rgy thorm stats that th rgy of th sigal may b writt i frqucy domai as suprpositio of rgis du to idividual Spctral frqucy of th sigal. i., E = x(t) dt = X ( ) d 6. Writ ay two proprtis of Fourir sris. Liarity Z(t) = a x(t) + b y(t) Fourir coff [Z(t)] = a C +b D. Whr C Fourir coff.of x(t) D Fourir coff.of y(t). Tim Shiftig : Fourir coff.of x(t-t 0 ) = - t0 T C. Whr C Fourir coff.of x(t). 7. Writ ay two proprtis of Fourir Trasform. Liarity: Ax (t) + Bx (t) FT AX ( ) + BX ( ) Whr X( X ( ) is Fourir trasform of x(t) ) is Fourir trasform of x (t). Tim shiftig: X(t t 0 ) FT - t 0 X( ). Whr X( ) is Fourir trasform of x(t).

18 8. Stat th Covolutio proprty of Fourir trasforms. Covolutio proprty says that covolutio i tim domai is quivalt to multiplicatio i frqucy domai X ( ) X ( ) X ( ).X( ) FT FT FT x (t) x (t). x (t) * x (t). Whr * dots covolutio..dots multiplicatio. 9. Stat ad prov tim diffrtiatio proprty of Fourir trasform. F [ x(t)] = ( ). Proof: Ivrs trasform quatio is giv by, x(t) = / X ( ) t d Diffrtiat both sids with rspct to t d/dt x(t) = / X ( ) t d = / ( ) t d Hc provd. = F - [ ( )] 0. Stat ad prov frqucy diffrtiatio proprty of Fourir trasform. F[ t x(t) ] = Proof: Fourir trasform is giv by

19 X( = x(t) - t Diffrtiat both sids with rspct to = F.T [ t x(t) ] Hc provd. Dfi frqucy domai trasfr fuctio. Trasfr fuctio of a LTI systm is th ratio of output i frqucy domai to iput i frqucy domai, which rprstd by. If is th Fourir trasform of th sigal x(t), th what is th Fourir trasform of F.T i trms of? Put

20 3. Th iput output of a causal LTI systm ar rlatd by th diffrtial quatio Frqucy rspos is H(w)= Giv: +6 +8y(t)=x(t) Takig Fourir trasform o both sids (w) y(w)+6wy(w)+8y(w)=x(w) Y(w)[(w) +6w+8]=x(w) H(w)= = 4. Fid Fourir sris rprstatio of th sigal x(t)=3cos( ). Giv, x(t)=3cos( ) x(t)=3[ ] x(t)= + Expotial Fourir sris is x(t)= O comparig, = => = =>t= = 4 Thrfor,c = c - = 5. Dtrmi ivrs Fourir trasform of X(w)= Giv, X(w)= dw Substitutig, X(w)= i th ivrs Fourir trasform quatio, w gt

21 x( t) ( w) wt dw By usig proprty of impuls fuctio ( t ) x( t) dt x(0) x (t) 6. What is th rlatio btw Fourir cofficits of trigoomtric ad xpotial form? C 0 =a 0 C 0 = [a -b ] Whr c 0,c xpotial Fourir sris cofficits. a0, a, b trigoomtric Fourir sris cofficit PART-B. Fid trigoomtric Fourir sris for th followig priodic sigal show i figur. Hit: odd sigal, so fid oly b. b = T T x( t)cos tdt As: x(t)= si( t)- si( t)+ 3 si(3 t)-.... Comput xpotial Fourir sris of th followig sigal show i th figur. Hit: Calculat tim priod T. Fid C o & C usig formula.

22 As: x(t)= [ -+ (-) ] 3. Dtrmi trigoomtric Fourir sris of full wav rctifid si wav. Hit: As this is v sigal, fid a o & a oly. As: x(t) = - [ ] 4. Fid th Fourir trasform of th sigal show i fig. Hit: Us Fourir trasforms formula As: X( )= T =T ). 5. Dtrmi th Fourir trasform of followig CT sigals.(i) (ii) x(t)= cos t. As: Writ th Fourir trasform quatio X = dt. (i). -.

23 (ii). 6. Fid ivrs Fourir trasform of x( )= ( - o) As:/ ot 7. Fid th covolutio of th sigals x (t)= -at u(t), x = -bt u(t). Us Fourir trasform. Hit: (i) Fid Fourir trasforms of x (t) i, x ( ) (ii) Fid Fourir trasform of x (t) i, x ( ) (iii)multiply x ( ) ad x ( ) (iv) Tak ivrs Fourir trasforms of rsult. 8. Th iput of causal LIT systm ar rlatd by diffrtial quatio d y(t)/dt +6dy(t)/dt+8y(t)=x(t). (a) Fid impuls rspos of th systm. (b) What is th rspos of th systm if x(t)=t -t u(t). Hit:.Fid trasfr fuctio H( )=y( )/x( ) by takig Fourir trasform..by takig ivrs Fourir trasforms of H( ),dtrmi impuls rspos h(t). 3.TakFourir trasforms of x(t) i, x( ). 4.Multiply x( ) ad H( ) i, y( ). 5.Tak ivrs Fourir trasform to gt th output i tim domai. As: h(t)= -t u(t)- -4t u(t) y(t)=/4 -t u(t)-/t -t u(t)+t / -t u(t)-/4-4t u(t).

24 UNIT-3 SAMPLING AND RECONSTRUCTION OF SIGNALS PART-A. Why CT sigal ar rprstd by sampls? (i) A CT sigal caot b rprstd i digital procssor or computr. (ii) To abl digital trasmissio of CT sigals.. What is mat by samplig? A Samplig is a procss by which a CT sigal is covrtd ito a squc of discrt sampls with ach sampl rprstig th amplitud of th sigal at th particular istat of tim. 3. Stat Samplig thorm. A bad limitd sigal of fiit rgy, which has o frqucy compots highr tha W HZ,is compltly rprstd by spcifyig th valu of th sigal at th istat of tim sparatd by /W scods. A bad limitd sigal of fiit rgy, which has o frqucy compots highr tha W HZ,is compltly rcovrd from th kowldg of its sampls tak at th rat of W sampls pr sc. 4. What is mat by aliasig? Wh high frqucy itrfrs with low frqucy ad appars as low frqucy, th th phomo is calld as aliasig. 5. What ar th ffcts of aliasig? Sic th high frqucy itrfrs with low frqucy th th distortio is gratd. Th data is lost ad it caot b rcovrd. 6. How th aliasig procss is limiatd? (i) Th samplig rat must b atlast twic th highst frqucy of origial sigal. fs W. (ii) Th sigal should b strictly bad limitd to W HZ. This ca b obtaid by usig low pass filtr bfor samplig procss. It is also calld as ati-aliasig filtr. 7. Dfi Nyquist rat ad Nyquist itrval. Wh samplig rat bcoms xactly qual to W sampls/sc, for a giv badwidth of W HZ, th it is calld Nyquist rat.

25 Nyquist Rat =W HZ Nyquist Itrval =/w sc 8. Dfi samplig of badpass sigals. A badpass sigal x(t) whos maximum badwidth is w ca b compltly rprstd ito ad rcovrd from its sampls, if it is sampld at th miimum rat of twic th badwidth. 9. What is a ati-aliasig filtr? To avoid aliasig rror, th iput sigal should b bad limitd to W HZ. So th iput sigal is bad limitd by usig a low pass filtr bfor samplig procss. Th lowpass filtr is calld as ati-aliasig filtr. 0. What ar th stps ivolvd i covrtig aalog sigal to digital sigal? (i) Aalog sigal is sampld by usig cotiuous tim sigal to discrt tim sigal. (ii) Th amplitud of th sigal is quatid. (iii)quatid sigal is covrtd to digital sigal.. What is ro ordr hold? How ca it b usd as a rcostructio filtr? This typ of filtr is o of th basic typ of rcostructio filtr. It simply holds th valu that is i fs() for τ scods. This crats a block or stp fuctio whr ach valu of th puls i fs() is simply draggd ovr to th xt puls.. A ral valud sigal x(t) is kow to b uiquly dtrmid by its sampls wh th samplig frqucy is s =0000π. For what valus of is X(w) guaratd to b ro? s m accordig to samplig thorm s-> samplig frqucy, m-> maximum frqucy m s/ m 0000π/ 5000π

26 3. A CT sigal x(t) is obtaid at th output of a idal lowpass filtr with cut off frqucy c=000π. If impuls trai samplig is prformd o x(t) which of th followig samplig priod would guarat that x(t) ca b rcovrd from its sampld vrsio usig a appropriat low pass filtr? (a) T= (b) T= 0 3 (c) T= 0 4 Giv maximum frqucy m =000π. So th rquird samplig frqucy is s m s s 000π 000π Or π fs 000 π f s H / Ts 000 Ts / sc. So th optio(a) ad (c) 0-4 satisfis th abov coditio. So it is possibl to rcovr origial sigal. But optio (b) 0-3 is gratr tha rquird samplig itrval which should ot b usd. 4. Th frqucy which, udr samplig thorm, must b xcdd by th samplig frqucy is calld Nyquist rat. Dtrmi th Nyquist rat corrspodig to ach of th followig sigals. (a) x(t)=+cos(000 t)+si(4000 t)

27 si(4000t) (b)x(t)= t (c)x(t)= si( 4000 t) t Nyquist rat f s =f m or ω s =ω m (a) Giv x(t)=+cos(000πt)+si(4000πt) Max.frqucy is ω m= 4000π πf m =4000π f m= h so th rquird Nyquist rat f s =f m f s =4000h si(4000 t) (b) Giv x(t)= t Max frqucy is ω m =4000π f m =000h so Nyquist rat is f s =4000h si( 4000 t) (c) giv x(t)= t cos(8000 t) x(t)= t max frqucy is ω m =8000π f m =4000h πf m =8000π So Nyquist rat is f s =f m F s =8000h. 5. What is mat by itrpolatio? Itrpolatio is usd to match th two adact samplig amplitud. Itrpolatio is isrtig som amplitud valus i btw two succssiv sampld poits. It is usd i rcostructig origial sigal from its sampld vrsio.

28 6. What is mat by aprtur ffct i samplig? How ca it b rducd? Aprtur ffct rsults from th fact that th sampl is obtaid as tim avrag with i a samplig rgio, rathr ust big qual to th sigal valu at th samplig istat.this is also calld as itgratio ffct.i a capacitor basd sampl ad hold circuit, th itgratio ffct is itroducd bcaus th capacitor caot istatly chag voltag thus rquirig th sampl to hav o-ro width. 7. What is mat by quatiatio? Whr it is usd? Quatiatio is th procss of mappig larg st of iput valus to a smallr st, such as roudig valus to som uit of prcisio. A dvic that prforms quatiatio is calld as quatir.th rror itroducd by quatiatio is rfrrd to as quatiatio is ivolvd i all digital sigal procssig opratio. 8. Dfi ovrsamplig ad udr samplig. Wh o sampl badpass sigal at a rat lowr tha th Nyquist rat, th sampls ar qual to sampls of low-frqucy alias of high frqucy sigal. This is calld as udr samplig. Ovr samplig mas th sigals ar sampld wll abov th Nyquist rat. This is usd i aalog to digital covrtor to rduc th distortio. 9. What ar diffrt mthods of samplig? Idal samplig (or) impuls trai samplig Natural samplig Flattop samplig 0. Th sigal x(t)=0 cosπt is idally sampld at a frqucy fs=00 sampls pr scod.sktch th spctrum of x s (t). X(f)=f[x(t)]=f[0 cos 50πt]

29 =5[δ(f-750+δ(f+75)] Sic th spctrumof xδ (t) is xδ(f) Xδ (f) =fs. For th x(t)=0 cos 50 πt, sktch xδ (f), th spctrum of x δ(t), th idally sampld vrsio of x(t), if th samplig is do at a frqucy fs=00sps.. How may miimum umbr of sampls ar rquird to xactly dscrib th followig sigals? x(t) = 0 cos (6πt)+4 si(8πt). T=priod of 0 cos (6πt) = = T=priod of 4 si (8πt) = =. = *4 = which is ratioal umbr. Hc x (t) is priodic T= LCM (, ) = Max frqucy prst i x(t) is 4h. Miimum samplig frqucy rquird is fs= 8 sps. 3. Dtrmi th miimum samplig frqucy to b usd to sampl th sigal x (t) =00 si 00t, if th sigal x(t) is to b rcivd from th sampls without ay distortio.

30 x (t) =00 si 00t = (0 si c 00t) (0sic 00t) W kow that 0sic 00tft 0.π (f/00) FT [(0 sic 00t ] = 0.π (f/00) * 0. π (f(00) = 0.0*00A (f/00) Hc Nyquist rat fs=00 sps. 4. Th sigal x(t)=cos(00πt)+6cos(80πt)is sampld at a frqucy of 50 sampls pr scod. Th sampld vrsio xδ(t) is passd through a uit gai idal LPF with a cut off frqucy compots will b prst i th output of LPF? Writ dow its output sigal. x (t) =cos (00πt) + 6cos (80πt) =cos πt(00)t + 6cos π(90 X (f) = [δ(f+00) + δ(f-00)] + 3[δ (f+90) + δ (f-90)] Th spctrum x δ (t) is idally sampld vrsio of x(t)

31 So th output cotais x(t)= [cos (50πt)+cosπ(00t]+6[cosπ(60)t+cos π(90)t] 5. Dtrmi Nyquist rat of samplig for followig sigals. (a) x(t)=0cos (00πt) (b) x(t)=0sic 00t (a) Max.frq is x(t)= (+cos00πt) ω m =00π ω s = ω m= 400π (b) Max.frq is 00 F s =f m =00ps. PART-B. Cosidr th samplig of a bad pass sigal whos spctrum is giv i figur. Dtrmi th miimum samplig rat f s to avoid aliasig. Hit: us samplig frqucy formula f s f m Fid max frqucy from th giv figur As:f s 4h. Cosidr a sigal x(t)=cos 000πt+0000πt+0cos 5000πt. Dtrmi th a) Nyquist rat for this sigal b) If samplig rat is 5000 sampls/sc, th what is discrt tim sigal obtaid aftr samplig. Hit: a) Nyquist rat is fs=fm

32 Fm=max.frqucy. b) Chck whthr 5000 sampls/sc is gratr tha Nyquist rat. 3. Prov samplig thorm with cssary rlatios xplai how origial sigal ca b rcovrd from its sampld vrsio. Hit: (i) Writ samplig thorm (ii) Cosidr a sigal with maximum frqucy of W H. (iii) Draw th spctrum bfor ad aftr samplig procss. (iv) Explai aliasig rror. (v) Driv th rcostructio filtr trasfr fuctio. 4. Wh a sc is shot with a movi camra, 30 frams of th sc ar grabbd by it vry scod. It is foud that somtims th rotatig whl of a motorbik i a movi sc appars to b rotatig i th backward dirctio, or ot rotatig at all. Explai this phomo. Hit: As Aliasig rror, th whl is lookig lik rotatig i backward dirctio. Explai aliasig ffct. 5. Th sigal y(t) is gratd by covolvig a bad limitd sigal x(t) with aothr bad limitd sigal x(t) that is y(t) = x(t)*x(t) whr X(ω) = 0 for ω >000π. X(ω) = 0 for ω >000π. Impuls trai samplig is prformd o y(t) to obtai y ( t) y( T ) ( t T ). Spcify th rag of p valus for samplig priod T which surs that y(t) is rcovrabls from yp(t). Hit: Explai impuls trai samplig mthod i dtail. 6. Explai samplig ad rcostructio of idal samplig mthod. Hit: (i) Writ samplig thorm. (ii) Draw th tim domai rprstatio of CT sigal, sampld sigal. (iii)aftr drivig frqucy domai rprstatio, draw th spctrum of sampld sigal.

33 UNIT-4 DISCRETE TIME SIGNALS AND SYSTEMS PART-A. Dfi LTI-CT Systms I a cotiuous tim systm if th tim shift i th iput sigal rsults i corrspodig tim shift i th output ad th systm obys th supr positio pricipl th th systm is calld LTI-CT systm.. What ar th tools usd for aalysis of DT-LTI Systms? (i) Discrt Tim Fourir Trasform(DTFT) (ii) Discrt Fourir Trasform(DFT) (iii)z-trasform. 3. Dfi uit stp, uit ramp ad uit impuls fuctio for DT sigals. Uit stp fuctio u( ) for 0 for 0 0 Uit Ramp fuctio r ( ) for 0 for 0 0

34 Uit impuls fuctio ( ) for 0 for Dfi LTI-DT Systms. A DT systm which has both liarity ad Tim ivariac proprty is calld as Liar Tim Ivariat (LTI)-Discrt Tim (DT) Systm. 5. Dfi DTFT. Lt us cosidr th discrt tim sigal x().its DTFT is dotd as X( ). It is giv as X ( ) x( ) 6. Stat th coditio for xistc of DTFT? Th coditios ar (i) Th sigal should b absolutly summabl i., x ( ) (ii) If x() is ot absolutly summabl, th it should hav fiit rgy for DTFT to xist. i., x( ) 7. List th proprtis of DTFT. Priodicity Liarity

35 Tim shiftig Frqucy shiftig Tim scalig Diffrtiatio is frqucy domai Tim rvrsal Covolutio Multiplicatio i tim domai Parsval s thorm 8. What is DTFT of uit sampl? DTFT quatio is X( w)= x ( ). -w Substitutig x()= () X( w )= ( ). -w X( w )=. 9. Dfi ivrs DTFT. x()=/ x( w ) w.d whr X( w ) DTFT of x(). 0. Dfi DFT DFT is dfid as X(k) which is giv by X(K)= N 0. Dfi Ivrs DFT. IDFT is dotd by x() which is giv by x( ). W N k for K 0 to N- x()=/n N K 0 X(k).w N -k for K 0 to N-

36 . Dfi Twiddl factor. - /N Th Twiddl factor is dfid as W N = 3. Dfi ro paddig. Th mthod of appdig ros i th giv squc is calld as ro paddig. 4. Dfi circularly v squc. A squc is said to b circularly v if it is symmtric about th poit ro o th circl. 5. Dfi circularly odd squc. X(N-)=x() for <=<=N- A squc is said to b circularly odd if it is ati-symmtric about th poit x(0) o th circl. 6. Dfi circularly foldd squc. A circularly foldd squc is rprstd as [x(-)] N. It is obtaid by plattig x() i clockwis dirctio alog th circl. 7. Stat circular covolutio. This proprty stats that multiplicatio of two DFT is qual to circular covolutio of thir squc i tim domai. 8. Stat Parsval s thorm. Cosidr th complx valud squcs x() ad y(). If x() DFT X(K) y() DFT Y(K) Th, x().y*() = /N K X(K).Y*(K).

37 9. What is mat by stp rspos of DT systm? Th output of th systm y() is obtaid for th uit stp iput u(), th it is said to b stp rspos of th systm. 0. Dfi frqucy rspos of DT systm. Frqucy rspos of DT systm is dotd by H( w )= Y( w) /X( w ) whr Y( w )= DTFT [y()] X( w )= DTFT[x()] y()= output of th systm x()= iput to th systm. Dfi impuls rspos of DT systm. Th impuls rspos is th output producd by DT systm wh uit impuls at th iput. Th impuls rspos is dotd by h(). is applid. Stat th sigificac of diffrc quatio. Th iput ad output bhavior of th DT systm ca b charactrid with th hlp of liar costat cofficit diffrc quatio. 3. Writ th gral diffrc quatio for discrt tim systm. Th gral form of costat cofficit diffrc quatio is y()= - k a k y(-k)+ k 0 b k x(-k) Hr k is th ordr of th quatio. x() is th iput ad y() is th output. 4. What is th coditio for stabl systm? A LTI systm is stabl if its impuls rspos is absolutly summabl. (i.), h() <

38 5. What ar th blocks usd for block diagram rprstatio? Th block diagrams ar rprstd with th hlp of scalar multiplirs, addrs ad dlay lmt. 6. Stat th sigificac of block diagram rprstatio? Th LTI systms ar rprstd with th hlp of block diagrams. Th block diagram is th mor ffctiv way of systm dscriptio. Block diagrams idicat how idividual calculatios ar prformd. 7. What ar th proprtis of covolutio? (i) Commutativ: x() * h() = h() * x() (ii) Associativ: x() * [h() * y()] = [x() * h()] * y() (iii)distributiv: x() * [y() + ()] = [x() * y()] + [x() * ()] 8. Dfi causal systm. For a LTI systm to b causal, h()=0 for <0. Wh h() impuls rspos of th systm. 9. What ar th classificatios of systm basd o uit sampl rspos? (i) FIR (Fiit Impuls Rspos) systm. (ii) IIR (Ifiit Impuls Rspos) systm. 30. Fid th Fourir trasform of x()={,,}. By dfiitio of Fourir trasform, X( w )= x( ) -w. = 0 x( ) -w. = + -w + -w = -w ( w + -w ) + -w

39 = -w. cosw + -w = -w [+ 4 cosw]. 3. Dfi th Fourir trasform of x() = u()- u(-n). x() ca b dfid as, x()= for =0 to N- X ( ) x( ) = N 0 = N 3. Fid th Fourir trasform of x() = a u( ) whr a <. By dfiitio of Fourir trasform, Giv x() = a u( ) X ( ) x( )

40 x() = 0 for for a ) ( a X = a ) ( a = ) ( 0 a = a = a a a X ) ( 33. Fid th DTFT of th sigal x() = ). ( (0.) ) (. 0 u u x X ) ( ) ( = = = 0 ) (0. 0.

41 = X ( ) cos 34. Dtrmi th ivrs Fourir trasform of X ) ( ). ( 0 Th ivrs Fourir trasform of X ( ) is x() = X ( ) d = ( 0 ) d x() = 0 PART B. Dtrmi th Discrt Fourir trasform of x() = {,,,,}. Hit: Us DFT dfiitio, X ( k) N 0 x( ) k/ N, whr k is ragig from 0 to N-. As: X(k) = {4,-.44,0,-0.44,0,+0.44,0,+.44}.. Fid th output squc y() if h() = {,,} ad x() = {,,3,} usig circular covolutio. Hit: (i) Appd ro i h() ad mak its lgth qual to 4. (ii) Rprst h() ad x() i circular form. (iii) Fid h(-). (iv) Usig circular covolutio formula, fid y(). y() = N m x ( m) y[( m 0 )] N

42 3. Lt x() ad h() b sigals with th followig Fourir trasforms. 3 3 ) ( X ad 4 ) ( H. Usig covolutio proprty of discrt tim Fourir trasform (DTFT), y() = x()*h(). Hit: (i) Multiply ) ( ) ( adh X. (ii) Tak ivrs DTFT for th rsult of (i). 4. Fid th discrt tim Fourir trasform of x() = ) ( u. Hit: Usig dfiitio of Fourir trasform x X ) ( ) ( As: 5. Cosidr a systm cosistig of th cascad of two LTI systms with frqucy rsposs H ) ( ad 4 ) ( H. Fid th diffrc quatio dscribig th ovrall systm. Hit: (i) As two systms coctd i cascad, multiply ) ( H ad ) ( H. Say ) ( H (ii) ) ( ) ( ) ( X Y H. So cross multiply ad tak ivrs Fourir trasform. As: ). ( ) ( 3) ( 8 ) ( x x y y 6. A causal LTI systm is dscribd by th diffrc quatio y()=y(-)+y(-)+x(-) (a) Fid th systm fuctio for this systm. (b) Fid th uit impuls rspos of th systm.

43 Hit: (i) Tak DTFT for th giv diffrc quatio (ii) Fid th systm fuctio H( (iii) Tak ivrs DTFT for th rsult (ii) to obtai impuls rspos. 7. Explai ay four proprtis of DTFT. Hit: (i) Liarity (ii) Tim shiftig (iii)tim rvrsd (iv) Tim scalig (v) Frqucy shiftig tc..

44 UNIT 5 FOURIER ANALSIS OF DT SIGNALS AND SYSTEMS PART A. Dfi covolutio sum. If x() ad h() ar discrt variabl fuctios, th its covolutio sum is giv by, y() = Whr x() Iput to th systm. h() Impuls rspos of th systm. k x ( k) h( k). List th stps ivolvd i fidig covolutio sum. Foldig Shiftig Multiplicatio Summatio 3. Dfi FIR systm. Th systm for which uit impuls rspos h() has fiit umbr of trms, thy ar calld as Fiit Impuls Rspos(FIR) systm. 4. Dfi IIR systm. Th systm for which uit impuls rspos h() has ifiit umbr of trms, thy ar calld as Ifiit Impuls Rspos(IIR) systm. 5. Dfi o-rcursiv ad rcursiv systm. Wh th output y()of th systm dpds upo prst ad past iputs oly th th systm is calld as o-rcursiv systm. Wh th output y() dpds upo prst ad past iputs as wll as past outputs, th it is calld as rcursiv systm. 6. Stat th rlatio btw Fourir trasform ad -trasform. X() = = X( ) at Z =.

45 7. Dfi systm fuctio. H(Z)= is calld systm fuctio (or) trasfr fuctio. It is Z-trasform of uit. Sampl rspos h() of th systm. 8. Dfi Z-trasform. Th Z-trasform of a discrt tim sigal x() is dotd by X() ad is giv by X()= x ( ) 9. Dfi ROC. Th valu of for which th -trasform covrgs is calld as Rgio of Covrgc(ROC). 0. Fid -trasform of x()={,,3,4}. x()={,,3,4} X()= x() as pr -traform dfiitio X()= 3 0 x() X()= Stat th covolutio proprty of - trasform. Th covolutio proprty stats that th covolutio of two squcs i tim domai is quivalt to multiplicatio of thir -trasforms. x()*y() x().y(). What is -trasform of x(-m)? By tim shiftig proprty,

46 Z[x(-m)]= x(-m) put -m=α =α+m [x(-m)]= x(α) = x(α) [x(-m)]= x() 3. Stat iitial valu thorm of -trasform. If x() is causal, th its iitial valu is giv by x(0) = 4. Stat Fial valu thorm of -trasform. x( )= (- )x() 5. List th mthods of obtaiig ivrs -trasform. Ivrs -trasform ca b obtaid by usig Partial fractio xpasio Cotour itgratio Powr sris xpasio Covolutio 6. What ar th proprtis of ROC? (i) Th ROC of a fiit duratio squc iclud tir -pla xcpt =0 ad =. (ii) ROC is always rig i th -pla ctrd about origi. (iii)roc dos ot cotai ay plas. (iv) ROC of casual squc is of th form >r. (v) ROC of lft hadd squc is of th form <r. (vi) ROC of two sidd squc is th coctric rig i th -pla.

47 7. What is diffrtiatio proprty of trasform? If x() is th squc. Th X() is its -trasform, x() X() x() - d d X(). 8. What is th coditio for causality if H() is giv? A discrt LTI systm with ratioal systm fuctio H() is casual if ad oly if (i) Th ROC is th xtrior of th circl outsid th outrmost pol. (ii) Wh H() is xprssd as a ratio of polyomials i Z, th ordr of th umrator caot b gratr tha th ordr of domiator. 9. What is th coditio for stability if H() is giv? A discrt LTI systm with ratioal systm fuctio H() is stabl if ad oly if all pols of H() lis isid th uit circl. That is thy must all hav magitud smallr tha. 0. Chck whthr th systm is casual or ot. Th H()= +/+. Th systm is ot casual, bcaus th ordr of umrator is gratr tha domiator.. Chck whthr th systm is stabl or ot. Giv H()= a whr a <. Th systm pol is -a=0. Z=a. As a < <. Sic all pols lis isid uit circl, th systm is stabl.. Dtrmi th trasfr fuctio for th systm dscribd by th diffrc quatio y()-y(-)=x()-x(-). By takig -trasform o both sids th trasfr fuctio y() - y() = x() - x()

48 y()[- ] = x()[- ] H()=y()/x() = - /- 3. Dfi raliatio structur. Th block diagram rprstatio of a diffrc quatio is calld ralid structur. Ths diagram idicat th mar i which th computatios ar prformd. 4. What ar diffrt typs of structur raliatio? (i) Dirct form (ii) Dirct form (iii)cascad form (iv) Paralll form 5. Dtrmi x(0) if th -trasform of x() is x() = /(+3)(+4). By iitial valu thorm of -trasform, x(0)=lt x() = lt /(+3)(+4) > > =lt / (+3/)(-4/) > =/(+0) = 6. Dtrmi th trasfr fuctio of LTI systm dfid by th quatio y()-0.5y(-)=x()+0.4x(-) Giv that y()-0.5y(-)=x()+0.4x(-) y()-0.5 y()=x()+0.4 x() y()[-0.5 ] = x()[+0.4 ] Trasfr f,h()=y()/x() = +0.4 / Th trasfr fuctio of LTI systm is H()=-/(-)(+3).Dtrmi th impuls rspos. H() =-/(-)(-3) = A/- + B/+3 -=A(+3)+B(-) Put = -3-4= -5B B=4/5 Put =

49 No.of complx multiplicatios=n =5 =5A A=/5 H() = (/5)/- + (4/5)/+3 H() = ((/5)/-) + ((4/5)/+3) h() =/5() - u(-)+4/5(-3) - u(-) 8. Dtrmi th stability ad causality of th systm dscribd by th trasfr fuctio, H()=/ /- for ROC:0.5< <. Giv that ROC is 0.5 < < Wh ROC is 0.5< <, th impuls rspos h() is two sidd sigal. Sic >0.5, th trm with pol =0.5 corrspods to right sidd sigal. Sic <, th trm with = corrspods to lft sidd sigal. Impuls rspos h()= {/ / } h() = 0.5 u() u( ). ) Th ROC icluds uit circl. Hc th systm is stabl. ) Th impuls rspos is two sidd sigal. Hc th systm is o-causal. 9. calculat th prctag savig i calculatios i a 5 poits radix FFT, wh compard to dirct DFT. Dirct computatio DFT o.of complx additios =N(N ) =5 5 =,6,63 =,6,44 Radix- FFT/o.of complx additios =N log N = 5 log 5 = 5 9 = 4608 No.of complx multiplicatios =N/ log N =5/ 9 = 304 Prctag savig i additio = 00 o.of additios i radix- FFT/o.of additios i dirct DFT 00 = 98.% Prctag savig i multiplicatio = =00 o.of multiplicatios radix- FFT/o.of multiplicatios i dirct DFT

50 =00 304/ =99.% 30. Compar DIT ad DIF radix- FFT. DIT radix- FFT DIF radix- FFT Tim domai squc is idpdc Frqucy domai squc is dcimatd. Th iput should b i bit rvrsd ordr; th Th iput should b i ormal ordr. Th output will b i ormal ordr. output will b i bit rvrsd ordr. I ach stag of computatios th phas factors I ach stag, th phas factors ar multiplid bfor add ad subtract opratios. multiplid aftr add ad subtract opratios 3. Arrag th 8-poit squc x()={,,3,4,-,-,-3,-4} i bit rvrsd ordr. x() i ormal ordr={,,3,4,-,-,-3,-4} x() i bit rvrsd ordr={,-,3,-3,,-,4,-4} 3. Why FFT is dd? Th FFT is dd to comput DFT with rducd umbr of calculatios. Th DFT is rquird for Spctrum aalysis ad filtrig opratios o th sigals usig digital computr. 33. What is bi spacig? Th N poit DFT of x() is giv by X(k) = N k/ N x ( ) = 0 N 0 x ( ) k W N Whr k W N k/ N phas is factor or twiddl factor. Th phas factors ar qually spacd aroud th uit circl at a frqucy icrmt of fs/n whr fs is th samplig frqucy of th tim domai sigal. This frqucy icrmt or rsolutio is calld bi spacig.

51 PART B. Dtrmi -trasform ad thir ROC of th followig discrt tim sigals. (a) x() = {3,,5,7} (b) x() = {6,4,5,3} (c) x() = {,4,5,7,3}. Hit: By dfiitio of -trasform X() = x ( ) As: (a) (b) (c) Dtrmi th -trasform ad thir ROC of th followig discrt tim sigals. (a) x() = u() (b) x() = 0.5 u( ) (c) x() = 0. u( ) (d) x() = 0.5 u( ) 0.8 u( ). Hit: by dfiitio of -trasform X() = x ( ) As: (a) ROC > (b) (c) > ROC <0. 0. (d) 0.5 ROC 0.5< <

52 3. Dtrmi th ivrs Z trasform of th fuctio x(z)= Z - -3Z - +Z - by partial fractio xpasio mthod ad powr sris xpasio mthod. Prov that th Ivrs -trasform is uiqu. HINT: Partial fractio xpasio X(Z)= 3Z +Z+/Z -3Z+ = 3Z +Z+/(Z-)(Z-) X(Z)/Z = 3Z +Z+/Z(Z-)(Z-) = A/Z + B/Z-+C/Z- Powr sris xpasio: Lt us divid umbr by domiator -3Z - +Z - 3+Z - +Z- ANS: x() = {3,,8,6,30.} 4. Dtrmi impuls rspos h() for th systm dscribd by th scod ordr diffrc quatio. y()-4y(-)+4y(-) = x(-) HINT: ) Tak Z-trasform o both sids ) Fid H() = y(z)/x(z) 3) Tak ivrs Z-trasform of H(Z) i; impuls rspos h() ANS: h() = - u(). 5. Dtrmi th rspos of LTI discrt tim systm govrd by th diffrc quatio by th diffrc quatio, y()-y(-)-3y(-) = x()+4x(-) for th iput x() = u() ad with iitial coditio y(-) =0 y(-) = 5 ANS: ) X(Z) = Z/ Z- ) FIND H(Z) 3) Y(Z) = X(Z). H(Z) 4) Tak ivrs Z-Trasform of y() y() = -4() u()+33/ (3) u()-3/(-) u().

53 6. Dtrmi th Fourir sris xpasio for th followig discrt tim sigals. (a) x() = cos 3 п (b) x() = 4 cosп/ (c) x() = 3 5п/ HINT: Usig Fourir Sris xpasio formula, C k = /N N k 0 x( ) -пk/n, K 0 to N- ANS: (a) X() is o-priodic.so Fourir sris dos ot xist. (b) x() = N k 0 c пk/n k = c 0 +c п/ + c п + c 3 3п/ = п/ + 3п/ 7. A 8poit squc is giv by X () = {,,,,,,, }. Comput 8poit DFT of x() by a) Radix- DITFFT ad b) Radix- DIF FFT. Also sktch magitud ad phas spctrum. Hit: X() i ormal ordr = {,,,,,,, } X() i bit rvrsal ordr = {,,,,,,, } Aswr: (a) Radix- DIT FFT: st stag output = {3,, 3,, 3,, 3, } d stag output = {6, -, 0, +, 6, -, 0, +} 3 rd stag output={, -.44, 0, -0.44, 0, +0.44, 0, +.44} (b) Radix- DIT DDT: st stag output = {3, 3, 3, 3,, , -, } d stag output = {6, 6, 0, 0, -, -.44, +, -.44} 3 rd stag output = {, 0, 0, 0, -.44, +0.44, -0.44, +.44} Fial output i both mthods, X (k) = {, -.44, 0, -0.44, 0, +0.44, 0, +.44}

54 B.E/B.TECH DEGREE EXAMINATION, NOVEMBER/DECEMBER 007 ANNA UNIVERSITY QUESTION R 3407 QUESTION WITH KEY ANSWERS PART A. What is th priodicity of x(t) = 00 t 30? 0 Giv x(t)= 00πt+30 Gral form of xpotial sigal is ωt+θ O comparig, w gt T 00 T = Draw th wavforms δ(t-) ad u(t+). 3. Dfi Fourir sris ad Fourir trasform. Fourir sris: x(t) = a 0 [ a cos t b si t] Whr aₒ, a, b ar Fourir cofficits

55 a 0 x( t) dt T T a T x( t)cos 0 T tdt b T x( t)si 0 T tdt Fourir trasform Fourir trasform is dfid by x( ) t x( ) x( t) dt for all 4. What is th diffrc btw Fourir trasform ad Laplac trasform? Fourir trasform is giv by x x( t) t dt Laplac trasform is giv by x( s) x( t) st dt Whr S is complx frqucy, S= Both ar usd to aalysis CT sigals. 5. What is mat by impuls rspos? Th CT ad DT systm, which producs th output for th uit impuls as iput, is calld as impuls rspos of th systm. 6. What ar th basic stps ivolvd i covolutio itgral? Basic stps ivolvd i covolutio itgral, * Tim rvrsal * Tim shiftig

56 * Multiplicatio * Itgratio or summatio. 7. Dfi Discrt Tim Fourir Trasform (DTFT). ) DTFT is dfid by, x ( x( ) 8. Dfi -trasform. Z-trasform is dfid by, X ( ) x( ) 9. What is th -trasform of u() ad δ()? Z- trasform of u(), X()= x ( ) = 0 = - δ x( ) X()= ( ) 0. Draw th block diagram of stat variabl quatio. Stat quatio: Q(t)= AQ(t)+BX(t).

57 PART-B (a) (i) Explai ral xpotial ad complx xpotial sigal. Ral xpotial x(t)= A t Complx xpotial X(t)= A ft =A[ cos( ft ) si( ft )] (a) (ii) Fid th priodicity of th sigal x() = si 3 cos. Giv x()=si / 3 cos / N=w= / 3 / N /3 N 3 N=>w= / / N / N 4 N /N = ¾ => ratioal umbr so priodic. N = LCM(N,N) =LCM(3,4) =.

58 .(b) (i) Vrify whthr th systm y() = x ( ) is liar ad tim ivariat systm. y()=x () ay ()=ax () by () = bx () ay ()+by () = [ax()+bx()] => ay ()+by () = ax () +bx () => so th systm is o liar. Y(,k)= x (-k) => Y(-k) = x (-k) => ()= () so th systm is tim ivariat. (b) (ii) Dcompos th sigal x(t) show i figur, i trms of basic sigals such as dlta, stp ad ramp. Giv, x(t) = -u(t)+u(t-)+u(t)-u(t-)-u(t-)+u(t-3)-r(t-4). x(t) = u(t)+u(t-)-3u(t-)+u(t-3)-r(t-4).

59 . (a) (i) Fid th Fourir trasform of th sigal show i figur. X(w)= x( t) wt dt = T / T / A wt dt = A wt w T / T / = A wt / wt / w = A w si wt / X(w)= A / wsi wt /. (a) (ii) Usig Fourir trasform proprtis fid th Fourir trasform of th sigal show i figur. X(w)= T / 4 0 A wt dt 3T / 4 A wt dt T / 4 3T / 4 T A wt dt = A wt w T / 4 0 A wt w 3T / 4 T / 4 A wt w T 3T / 4

60 A = A wt / 4 w3t / 4 w w A w A w3t / 4 w A wt / 4 w A wt w wt / 4 w3t / 4 = A A A wt A w w w w. (b) (i) Fid th Laplac trasform of x(t) = t at u(t). X(s)= x( t) st dt X(s)= t at u( t) st dt X(s)= 0 t ( s a) t dt (s a) t u=t dv= dt u = v= ( s a) t ( s a) x(s)= t ( s a) t ( s a) 0 0 ( s a) t s a dt = ( s a) t ( s a) 0 ( s = a). (b) (ii) Fid th ivrs Laplac trasform of X(s) = ( s )( s. ) x(s)= ( s )( s ) = A s B s

61 = A ( s ) B( s ) Put s=- B B s A x( t) t u( t) t u( t) 3. (a) Fid th uit stp rspos of th circuit show i figur. From th figur, y( w) X ( w) R wl R Y(w)= X ( w) w 0 Y ( w) 5 wy ( w) 0X ( w) Giv, X(w)= / w Y( w) 0 5 w 0/ w Y(w)= 0 w(0 5 w) = A/ w B /(0 5 w ) Y(w) = A w 0 B 5 w 0 = A (0+5w) + Bw Put w=0 A=

62 Jw=- 0=-B B=-5 t y(t) = u(t) - u( t) 3. (b) Cosidr a LTI systm whos rspos to th iput x(t) = t 3t u( t) is y(t) = t 4t u( t). Fid th systm s impuls rspos. X(w) = w w 3 w ( w 3 w )( w 3) w 4 ( w )( w 3) Y(w) = w w 4 w ( w 8 w )( w 4) ( w 6 )( w 4) H(w) = y( w) x( w) ( w 6 )( w 4) ( w )( w ( w ) 3) = 3( w 3) ( w )( w 4) A w B w 4 3(w+3) =A (w+4) + B (w+) -3 = -B B= t t h(t) = [ ] u( t). 4 (a) (i) Fid th -trasform of x() = u( ) ad plot th pol-ro pattr. 4 x() = [( ) ( ) ] u( ) 4

63 X() = 4. (a) (ii) Stat ad prov tim shiftig proprty ad tim covolutio proprty of -trasform. Tim shiftig proprty x() x() x(-m) m x(). Tim covolutio proprty x() * y() x().y() 4.(b) (i) What is th diffrc btw Discrt Fourir trasform (DFT) ad Discrt Tim Fourir trasform(dtft)? Discrt Tim Fourir Trasform X ) ( ) x( ; By takig sampls, from x( w ), w gt Discrt Fourir trasform x(k)= N 0 x ( ) k N 4. (b) (ii) Fid th discrt Fourir trasform of x() = {,,, }. DFT of x() = x(k) = {4,0,0,0} 5. (a) Fid th output of th systm whos iput output is rlatd by th diffrc quatio 5 y( ) y( ) y( ) x( ) x( ) for th stp iput Y() - y( ) y( ) x( ) x( ) 6 6

64 5 Y() [- ] x( )[ ] 6 6 Y()= ( 5 6 ) 6 = ( ( )( ) )( 3 ) A B = Z 3 = A( Z ) B( ) 3 y()= u() ( u(). 5. (b) Draw th stat spac quatio of a gral discrt systm with a xampl. Stat spac quatio of gral systm. Lt (), (). () >N stat variabls (), (). () >M iputs (), (). () >P outputs

65 ) (.. ) ( ) ( q q q N = a a a a a a a a a ) (.. ) ( ) ( q q q N b b b b b b b b b.. ) ( ) ( ) ( 3 x x x A >Systm matrix B >Iput matrix Q() >Stat vctor x() >Iput vctor

66 B.E/B.TECH DEGREE EXAMINATION, NOVEMBER/DECEMBER 007 ANNA UNIVERSITY QUESTION V 4588 QUESTION WITH KEY ANSWERS PART-A. Giv x() = {,-,, 3 4}. Rprst x() as a liar combiatio of wightd shiftd impuls fuctios. x()= (+)- (+)+ ()+3 ()+4 (-). Dtrmi whthr th sigal x(t) = cos 00πt + 5 si 50t is priodic. Giv x(t)=cos00 t+5si50t is priodic or ot. T=> 00 => / T 00 T / 50 T=> 50 / T 50 T / 5 T/T =/50*5/ / irratioal umbr. So th giv sigal is o priodic. 3. If X(Ω) is th Fourir trasform of x(t), what is th Fourir trasform of x(t-) i trms of X(Ω). F.T[x(t-)]= x (t ) wt dt. Put t- = t dt d x( ) d X ( ) 5t 4. Fid th Laplac trasform of x( t) u( t ) ad spcify its rgio of covrgc. X(S)= 5 t u(t-) 5t dt

67 = ( s 5) t ( s 5) t dt = ( s 5 ) ( s 5) = s 5 5. Th iput output of a causal LTI systm ar rlatd by th diffrtial quatio d y( t) dy( t) 6 8y( t) dt dt x( t). Fid th frqucy rspos H(Ω) of th systm. ( ) Y( ) +6 Y ( ) X ( ) X ( ) Y( )[( ) 8] 6 =X( ) H( ) ( ) Stat Dirichlt s coditios. Dirichlt s coditio: (i) Th fuctio x(t) should b sigl valud i ay fiit tim itrval T (ii) Th fuctio x(t) should hav atmost fiit umbr of discotiuitis i ay fiit ti itrval (iii)th fuctio x(t) should hav fiit umbr of maxima ad miima i ay fiit ti itrval. (iv) Th fuctio x(t) should b absolut itgratd 7. Fid th Fourir trasform of x() = δ(+) δ(-). X()= x ( ) Z = ( )Z - ( )Z =Z - Z 8. Dfi rgio of covrgc of -trasform. Th valu of Z for which Z trasform covrgs is calld as ROC.

68 9. Fid -trasform of th squc x() = {,, 3, -} ad its ROC. X()=Z +Z+3-/Z ROC tir Z-plac xcpt Z=0,Z= 0. Giv th diffrc quatio y ( ) y( ) x( ). Draw th block diagram rprstatio. PART B. (a) (i) Giv x() = y()=x()+/y(-),,,,,,, Plot th followig sigals. ) x(3-) ) x(3) 3) x()u(3-)

69 . (a) (ii) Sktch th v ad odd part of th sigal x(t)..(b) (i) Giv y() = x(). Dtrmi whthr th systm is mmorylss, causal, liar ad tim variat. Systm is mmorylss, causal, Liar, Tim variat.. (b) (ii) Prov that th powr of rgy sigal is ro ovr ifiit tim ad rgy of powr sigal ifiit ovr ifiit tim. E = x( t) dt P = Lt T T T T x( t) dt. (a) Cosidr a causal LTI systm implmtd as th RL circuit show blow.

70 A currt sourc producs a iput currt x(t) ad th systm output is cosidrd to b th currt y(t) flowig through th iductor. (i) Fid th diffrtial quatio rlatig th x(t) ad y(t) (ii) Dtrmi th frqucy rspos of this systm by cosidrig th output of th systm to th iputs of th form x(t) = t. Y(w)=X(w) w Y(w)+w Y(w)=X(w) y(t)+ dy ( t) =x(t) dt Y( w). (ii) H(w) = = X ( w) w.(b)(i) Fid ivrs Laplac trasform of th fuctio 3s 7 X ( s) s s X ( s) ( s 3s 7 3)( s 3s 7 X ( s) s s 3 ) ROC: R(S)>3. 3 A B = s 3 s 3s+7 = A(s+)+B(s-3) Put s=- 4 = -4B => B = - s=3 6 = 4A => A= 4 x(t) = 4 3t u(t) - t u(t).(b)(ii) Stat ad prov th tim shiftig proprty of cotiuous tim Fourir trasform. Tim shiftig proprty x(t) x(t-t) FT FT X(w) wt (w). t 3.(a) Covolv th followig sigals for α β ad α=β. x( t) u( t) t ad h( t) u( t). x(τ) = u(τ) h(t-τ) = (t ) u(t-τ)

71 y(t) = x(τ) h(t-τ) dτ = t 0 dτ. = t 0 ( ) dτ = t ( ) = t ( ) 3 (b) Th iput output rlatio of a systm at iitial rst is giv by d y( t) dt dy( t) 4 dt 3y( t) dx( t) dt x( t) fuctio (ii) Frqucy rspos (iii) Impuls rspos. H(s)= H(w) = = A w Y( s) X `( s) s. Usig Laplac trasform, fid (i) systm trasfr s Y(s) + 4 sy (s) + 3y(s) = s x(s) + x(s) s 4s w ( w) 4 w B 3 w w+ = A(w+) + B(w+3) h(t) = ½ 3 3 3t u(t) + ½ t u(t) -> systm trasfr fuctio 4.(a) Lt x() ad h() b sigals with th followig Fourir trasforms. 3 X ( ) 3. 4 H ( ). Usig covolutio proprty of discrt tim Fourir trasform, dtrmi y() = x()*h(). X( w ).H( w )=-3 X( w )H( w ) = 3 w + 6 5w 4w w 3 5w w w 4w 3w 3w w 5w 4 3w y() = { 3,, -, -3,,, 6, 0, 0, -} w w w 3w 3 6. w

72 4 (b) (i) Fid th discrt tim Fourir trasform of x() = ( ). u X( w ) = w = = = w - = w w - 4 (b) (ii) Prov th diffrtiatio proprty of trasform. Diffrtiatio proprty of trasform x() x() w w w x() d - x() d 5 (a)(i) Covolv th followig discrt tim sigals usig covolutio sum formula. X() = h() = u(). y() = = 5 (a) (ii) Giv th diffrc quatio y() y(-) = x(). Fid th impuls rspos by solvig rcursivly. Y() - y()= x() H() = = h() = () u() 5(b) Usig trasform, solv th followig diffrc quatio 5 y( ) y( ) y( ) x( ) with x() =,.Assum ro iitial coditios Y() - - y() + - y() = x() 6 6

73 Y() = Y() = y() = ( = 3 )( 3 ) u() =,

74 B.E/B.TECH DEGREE EXAMINATION, NOVEMBER/DECEMBER 007 ANNA UNIVERSITY QUESTION L 075 QUESTION WITH KEY ANSWERS PART-A. Dfi causality ad stability of a systm with a xampl for ach. Causality is dfid aas th systm should dpds upo oly prst ad past iput but ot o futur iput. Ex. y()= x()+x(-) Stability is dfid as th systm which producs badd output for vry boudd iput. Ex. y()=x() + x(-). Stat Dirichlt s coditios for Fourir sris. Dirichlt coditios: (i) Th fuctio x(t) should hav sigl valu i ay fiit tim itrval T (ii) Th fuctio x(t) should hav fiit o. of discotiuitis i ay fiit tim itrval T (iii)th fuctio x(t) should hav fiit o. of maxima ad miima i ay fiit itrval. (iv) Th fuctio x(t) should b absolutly itgrabl 3. Dtrmi ivrs Fourir trasform of X(ω) = δ(ω). Ivrs Fourir trasform is giv by, x( t) x( w) = ( w) dw dw = by usig impuls proprty 4. Stat th coditios for causality ad stability of systm with impuls rspos h(). Causality: h()=0 <0 Stability: h ( )

75 5. What is mat by aliasig? How ca it b avoidd? High frqucy compots of a sigal poss as low frqucy compot. This is calld as aliasig. Aliasig rror occurs bcaus of udr samplig. To avoid aliasig rror, th sigal should b sampld at a frqucy gratr tha twic th highst frqucy of origial sigal. 6. Fid -trasform of th giv data squc, x() = ;0 0 0; othrwis. trasform x()= Stat th tim shiftig ad frqucy shiftig proprtis of DTFT. Tim shiftig proprty of DTFT: x() DTFT x( x(-k) DTFT Frqucy shiftig proprty of DTFT: x( ) X ( 0 DTFT ( 0 ) 8. List out th ways for itrcoctig ay two LTI systms. (i) Cascad coctio (ii) Paralll coctio 9. Distiguish btw IIR ad FIR systms. FIR Systm: Th output of th systm for impuls as iput will b fiit lgth IIR Systm: IIR systm producs th impuls rspos with ifiit lgth. 0. How may umbr of additios, multiplicatios ad mmory locatios ar rquird to rali a IIR systm with trasfr fuctio, H() havig M ros ad N pols i dirct form I ad dirct form II raliatios? )

76 Dirct Form I b b b H ( 0 ) a a b a No of additios: M+N- No of multiplicatios: M+N No of dlay lmts: M+N Dirct Form II No of additios: M+N- No of multiplicatios: M+N No of dlay lmts: M or N whichvr is highr PART B. (a) (i) Dtrmi th Fourir sris rprstatio of a priodic squar wav ad dfid ovr o priod x(t) = ; t 0; T T t T /. It is Ev sigal. So fid oly a 0 ad a. a 0 x( t) dt T T a x( t)cos 0 T T t dt

77 . (a) (ii) Tst whthr th followig sigals ar priodic or ot ad if th sigal is priodic calculat th fudamtal priod. (i) (ii) x() = cos( ) si( ) 3cos( ) x ( t) 3cos(5t ). 6 x ( t) 3cos(5t ). 6 5 =5 T T = 5 It is Priodic. x() = cos( ) si( ) 3cos( ) N N 4 N N 8 8 N 3 N N = LCM (N,N,N3) = 6. It is priodic.. (a) (i) Fid th Fourir trasform of th sigal x(t) = at u( t); a 0 ad plot th magitud ad phas spctrum. X ( ) a X ( ) a a

78 = a a a = a X ( ) ta a (a) (ii) Stat ad prov Parsval s rlatio for cotiuous tim sigals usig Fourir trasform. Parsval s Thorm: x( t) dt x( w) dw (b) (i) Obtai th output of th systm by covolutio, giv th impuls rspos h(t) = u(t) ad iput sigal x(t) = at u(t). x t at u T h t T u t T y(t)= t 0 at dt at a t 0 = at a t o = at (b)(ii) Fid th ivrs Laplac trasform of cas (i) - > R(s) > -4. Cas (ii) R(s) < X ( s) if ROC is ( s )( s 4)

79 A= B= - 4 = A(s+4) + B(s+) Cas (i): x(t)= t u(t) - 4t u(t) Cas (ii): x(t)= - t u(-t-) + 4t u(-t-) 3(a)(i) Stat tim shiftig ad frqucy shiftig proprtis of -trasform. Tim Shiftig x() x(z) x(-k) k x(z) Diffrtiatio i frqucy domai: x() d x() d 3 (a)(ii) Cosidr th samplig of bad pass sigal whos spctrum is giv i figur. Dtrmi th miimum samplig rat Fs to avoid aliasig. Miimum samplig rat fs=4 sps a 3 (a)(iii) Usig diffrtiatio proprty, dtrmi th ivrs trasform for x()= ( a ) ; >a. a x()= ; >a ( a )