CHAPTER 3 SIGNALS & SYSTEMS. z -transform in the z -plane will be (A) 1 (B) 1 (D) (C) . The unilateral Laplace transform of tf() (A) s (B) + + (D) (C)

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1 CHAPER SIGNALS & SYSEMS YEAR ONE MARK n n MCQ. If xn [ ] (/) (/) un [ ], hen he region of convergence (ROC) of i z ranform in he z plane will be (A) < z < (B) < z < (C) < z < (D) < z MCQ. he unilaeral Laplace ranform of f () i. he unilaeral Laplace + + ranform of f() i (A) (B) + ( + + ) ( + + ) (C) ( ) (D) ( ) + + YEAR WO MARKS MCQ. Le yn [ ] denoe he convoluion of hn [ ] and gn, [ ] where hn [ ] ( / ) n un [ ] and gn [ ] i a caual equence. If y [] and y [] /, hen g [] equal (A) (B) / (C) (D) / MCQ.4 he Fourier ranform of a ignal h () i Hj ( ω) ( co ω)( in ω)/ ω. he value of h() i (A) 4 / (B) / (C) (D) MCQ.5 he inpu x () and oupu y () of a yem are relaed a y () x( τ) co( τ) dτ. he yem i (A) imeinvarian and able (B) able and no imeinvarian (C) imeinvarian and no able (D) no imeinvarian and no able GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

2 PAGE 6 SIGNALS & SYSEMS CHAP YEAR ONE MARK MCQ.6 he Fourier erie expanion f () a + / an con ω + bninn ω of n he periodic ignal hown below will conain he following nonzero erm (A) a and b, n 5,,,... (B) a and a, n,,,... n n (C) a a n and bn, n,,,... (D) a and a n 5,,,... n MCQ.7 Given wo coninuou ime ignal x () e and y () e which exi for * >, he convoluion z () x () y () i (A) e e (B) e (C) e + (D) e + e YEAR WO MARKS MCQ.8 Le he Laplace ranform of a funcion f () which exi for > be F () and he Laplace ranform of i delayed verion f ( τ) be F (). Le F *( ) be he complex conjugae of F () wih he Laplace variable e σ+ jω. F() F*() If G (), hen he invere Laplace ranform of G () i an ideal F () (A) impule δ () (B) delayed impule δ( τ) (C) ep funcion u () (D) delayed ep funcion u ( τ) MCQ.9 he repone h () of a linear ime invarian yem o an impule δ (), under iniially relaxed condiion i h () e + e. he repone of hi yem for a uni ep inpu u () i (A) u () e + + e (B) ( e + e ) u( ) (C) ( 5. e 5. e ) u() (D) e δ () + e u() YEAR ONE MARK MCQ. For he yem /( + ), he approximae ime aken for a ep repone o reach 98% of he final value i (A) (B) (C) 4 (D) 8 GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

3 CHAP SIGNALS & SYSEMS PAGE 7 MCQ. he period of he ignal x ( ) 8 in.8 ` + 4 j i (A).4 (B).8 (C).5 (D).5 MCQ. MCQ. he yem repreened by he inpuoupu relaionhip 5 y () x( τ) dτ, > (A) Linear and caual (C) Caual bu no linear (B) Linear bu no caual (D) Neiher liner nor caual he econd harmonic componen of he periodic waveform given in he figure ha an ampliude of (A) (B) (C) / (D) 5 YEAR WO MARKS MCQ.4 x () i a poiive recangular pule from o + wih uni heigh a hown in he figure. he value of X( ω) dω " where X( ω) i he Fourier ranform of x ()} i. (A) (C) 4 (B) (D) 4 MCQ.5 Given he finie lengh inpu xn [ ] and he correponding finie lengh oupu yn [ ] of an LI yem a hown below, he impule repone hn [ ] of he yem i GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

4 PAGE 8 SIGNALS & SYSEMS CHAP (A) hn [ ] {,,, } (B) hn [ ] {,, } (C) hn [ ] {,,, } (D) hn [ ] {,, } Common Daa Queion Q.67. Given f () and g () a how below MCQ.6 g () can be expreed a (A) g () f( ) (B) g () f ` j (C) g () f ` j (D) g () f ` j MCQ.7 he Laplace ranform of g () i (A) ( ) e e 5 (B) ( ) e 5 e (C) e ( e ) 5 (D) ( ) e e YEAR 9 ONE MARK MCQ.8 A Linear ime Invarian yem wih an impule repone h () produce oupu y () when inpu x () i applied. When he inpu x ( τ) i applied o a yem wih impule repone h ( τ), he oupu will be (A) y() τ (B) y( ( τ)) (C) y ( τ) (D) y ( τ) YEAR 9 WO MARKS MCQ.9 A cacade of hree Linear ime Invarian yem i caual and unable. From hi, we conclude ha (A) each yem in he cacade i individually caual and unable (B) a lea on yem i unable and a lea one yem i caual (C) a lea one yem i caual and all yem are unable GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

5 CHAP SIGNALS & SYSEMS PAGE 9 (D) he majoriy are unable and he majoriy are caual MCQ. he Fourier Serie coefficien of a periodic ignal x () expreed a jk/ x () / ae k are given by a j, a.5 + j., a j, k a 5. j., a + j and ak for k > Which of he following i rue? (A) x () ha finie energy becaue only finiely many coefficien are nonzero (B) x () ha zero average value becaue i i periodic (C) he imaginary par of x () i conan (D) he real par of x () i even MCQ. he zranform of a ignal xn [ ] i given by 4z + z + 6z + z I i applied o a yem, wih a ranfer funcion Hz () z Le he oupu be yn. [ ] Which of he following i rue? (A) yn [ ] i non caual wih finie uppor (B) yn [ ] i caual wih infinie uppor (C) yn [ ] ; n > (D) Re[ Yz ( )] ji z e Re[ Yz ( )] ji z e Im[ Yz ( )] ji Im[ Yz ( )] j i; θ < z e z e YEAR 8 ONE MARK MCQ. he impule repone of a caual linear imeinvarian yem i given a h. () Now conider he following wo aemen : Saemen (I): Principle of uperpoiion hold Saemen (II): h () for < Which one of he following aemen i correc? (A) Saemen (I) i correc and aemen (II) i wrong (B) Saemen (II) i correc and aemen (I) i wrong (C) Boh Saemen (I) and Saemen (II) are wrong (D) Boh Saemen (I) and Saemen (II) are correc α MCQ. A ignal e in( ω) i he inpu o a real Linear ime Invarian yem. Given K and φ are conan, he oupu of he yem will be of he form β Ke in( v + φ) where (A) β need no be equal o α bu v equal o ω (B) v need no be equal o ω bu β equal o α (C) β equal o α and v equal o ω (D) β need no be equal o α and v need no be equal o ω GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

6 PAGE SIGNALS & SYSEMS CHAP YEAR 8 WO MARKS MCQ.4 A yem wih x () and oupu y () i defined by he inpuoupu relaion : y () xd () τ he yem will be (A) Caual, imeinvarian and unable (B) Caual, imeinvarian and able (C) noncaual, imeinvarian and unable (D) noncaual, imevarian and unable MCQ.5 A ignal x () inc( α) where α i a real conan ^inc() x h i he inpu o a Linear ime Invarian yem whoe impule repone h () inc( β), where β i a real conan. If min ( αβ, ) denoe he minimum of α and β and imilarly, max ( αβ, ) denoe he maximum of α and β, and K i a conan, which one of he following aemen i rue abou he oupu of he yem? (A) I will be of he form Kinc( γ ) where γ min( α, β) (B) I will be of he form Kinc( γ ) where γ max( α, β) (C) I will be of he form Kinc( α) (D) I can no be a inc ype of ignal GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN: in( x) x MCQ.6 Le x () be a periodic ignal wih ime period, Le y ( ) x ( ) + x ( + ) for ome. he Fourier Serie coefficien of y () are denoed by b k. If bk for all odd k, hen can be equal o (A) / 8 (B) / 4 (C) / (D) MCQ.7 Hz () i a ranfer funcion of a real yem. When a ignal xn [ ] ( + ) i he inpu o uch a yem, he oupu i zero. Furher, he Region of convergence (ROC) of ^ z h H(z) i he enire Zplane (excep z ). I can hen be inferred ha Hz () can have a minimum of (A) one pole and one zero (B) one pole and wo zero (C) wo pole and one zero D) wo pole and wo zero MCQ.8 Given Xz () z wih z > a, he reidue of Xzz () n a z a for ( z a) n $ will be (A) a n (B) a n (C) na n (D) na n j n

7 CHAP SIGNALS & SYSEMS PAGE MCQ.9 Le x () rec^ h (where rec() x for x and zero oherwie. in( x) If inc() x x, hen he Fof x () + x( ) will be given by (A) inc ` ω j (B) inc ω ` j (C) inc ω co ω ` j ` j (D) inc ω in ω ` j ` j MCQ. Given a equence xn, [ ] o generae he equence yn [ ] x[ 4n], which one of he following procedure would be correc? (A) Fir delay xn ( ) by ample o generae z [ n], hen pick every 4 h ample of z [ n] o generae z [ n], and han finally ime revere z [ n] o obain yn. [ ] (B) Fir advance xn [ ] by ample o generae z [ n], hen pick every 4 h ample of z [ n] o generae z [ n], and hen finally ime revere z [ n] o obain yn [ ] (C) Fir pick every fourh ample of xn [ ] o generae v [ n], imerevere v [ n] o obain v [ n], and finally advance v [ n] by ample o obain yn [ ] (D) Fir pick every fourh ample of xn [ ] o generae v [ n], imerevere v [ n] o obain v [ n], and finally delay v [ n] by ample o obain yn [ ] YEAR 7 ONE MARK MCQ. Le a ignal ain( ω+ φ) be applied o a able linear ime varian yem. Le he correponding eady ae oupu be repreened a af ( ω+ φ). hen which of he following aemen i rue? (A) F i no necearily a Sine or Coine funcion bu mu be periodic wih ω ω. (B) F mu be a Sine or Coine funcion wih a a (C) F mu be a Sine funcion wih ω ω and φ φ (D) F mu be a Sine or Coine funcion wih ω ω MCQ. he frequency pecrum of a ignal i hown in he figure. If hi i ideally ampled a inerval of m, hen he frequency pecrum of he ampled ignal will be GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

8 PAGE SIGNALS & SYSEMS CHAP YEAR 7 WO MARKS MCQ. A ignal x () i given by, / 4 < / 4 x () *, / 4< 7/ 4 x ( + ) Which among he following give he fundamenal fourier erm of x ()? (A) 4 co ` 4 j (B) co 4 ` + 4j (C) 4 in ` 4 j (D) in 4 ` + 4j Saemen for Linked Anwer Queion 4 and 5 : MCQ.4 A ignal i proceed by a caual filer wih ranfer funcion G () For a diorion free oupu ignal wave form, G () mu (A) provide zero phae hif for all frequency (B) provide conan phae hif for all frequency GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

9 CHAP SIGNALS & SYSEMS PAGE (C) provide linear phae hif ha i proporional o frequency (D) provide a phae hif ha i inverely proporional o frequency MCQ.5 Gz () αz + βz i a low pa digial filer wih a phae characeriic ame a ha of he above queion if (A) α β (B) α β ( / ) (C) α β (D) α ( / ) β MCQ.6 Conider he dicreeime yem hown in he figure where he impule repone of Gz () i g(), g() g(), g() g(4) g hi yem i able for range of value of K (A) [, ] (B) [, ] (C) [, ] (D) [, ] MCQ.7 If u (), r () denoe he uni ep and uni ramp funcion repecively and u ()* r () heir convoluion, hen he funcion u ( + )* r ( ) i given by (A) ( ) u( ) (B) ( ) u ( ) (C) ( ) u( ) (D) None of he above MCQ.8 Xz () z, Yz () + z are Z ranform of wo ignal xn [ ], yn [ ] repecively. A linear ime invarian yem ha he impule repone hn [ ] defined by hee wo ignal a hn [ ] xn [ ]* yn [ ] where * denoe dicree ime convoluion. hen he oupu of he yem for he inpu δ[ n ] (A) ha Zranform z X() z Y() z (B) equal δ[ n] δ[ n ] + δ[ n4] 6δ[ n5] (C) ha Zranform z + z 6z (D) doe no aify any of he above hree YEAR 6 ONE MARK MCQ.9 he following i rue (A) A finie ignal i alway bounded (B) A bounded ignal alway poee finie energy (C) A bounded ignal i alway zero ouide he inerval [, ] for ome (D) A bounded ignal i alway finie GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

10 PAGE 4 SIGNALS & SYSEMS CHAP MCQ.4 x () i a real valued funcion of a real variable wih period. I rigonomeric Fourier Serie expanion conain no erm of frequency ω ( k)/ ; k,g Alo, no ine erm are preen. hen x () aifie he equaion (A) x () x ( ) (B) x () x ( ) x( ) (C) x ( ) x ( ) x ( / ) (D) x ( ) x ( ) x ( / ) MCQ.4 A dicree real all pa yem ha a pole a z + % : i, herefore (A) alo ha a pole a + % (B) ha a conan phae repone over he z plane: arg Hz () conan conan (C) i able only if i i anicaual iω (D) ha a conan phae repone over he uni circle: arg He ( ) conan YEAR 6 WO MARKS MCQ.4 xn [ ] ; n<, n>, x[ ], x[ ] i he inpu and yn [ ] ; n<, n>, y[ ] y[], y[], y[] i he oupu of a dicreeime LI yem. he yem impule repone hn [ ] will be (A) h[ n] ; n <, n >, h[ ], h[ ] h[ ] (B) hn [ ] ; n<, n>, h[ ], h[ ] h[ ] (C) h[ n] ; n <, n >, h[], h[], h[] (D) hn [ ] ; n<, n>, h[ ] h[ ] h[ ] h[ ] MCQ.4 he dicreeime ignal xn [ ] Xz ( ) n n z n /, where n + denoe a ranformpair relaionhip, i orhogonal o he ignal (A) y [ n] Y( z) n n ) / z n `j n (B) y [ n] ) Y ( z) ( 5 n) z / / (C) y [ n] ) Y ( z) z n n n n (D) y [ n] ) Y ( z) z + z ( n + ) MCQ.44 A coninuouime yem i decribed by y () e x (), where y () i he oupu and x () i he inpu. y () i bounded (A) only when x () i bounded (B) only when x () i nonnegaive GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

11 CHAP SIGNALS & SYSEMS PAGE 5 (C) only for if x () i bounded for $ (D) even when x () i no bounded MCQ.45 he running inegraion, given by y () xd () (A) ha no finie ingulariie in i double ided Laplace ranform Y () (B) produce a bounded oupu for every caual bounded inpu (C) produce a bounded oupu for every anicaual bounded inpu (D) ha no finie zeroe in i double ided Laplace ranform Y () ' ' YEAR 5 WO MARKS MCQ.46 For he riangular wave from hown in he figure, he RMS value of he volage i equal o (A) (C) 6 MCQ.47 he Laplace ranform of a funcion f () i F () ", f( ) approache (A) (B) 5 (C) 7 (D) (B) (D) MCQ.48 he Fourier erie for he funcion fx () inxi (A) in x+ in x (B) co x (C) in x+ co x (D) cox ( + + ) a MCQ.49 If u () i he uni ep and δ () i he uni impule funcion, he invere z ranform of Fz () z + for k > i (A) ( ) k δ( k) (B) δ() k ( ) k (C) ( ) k uk ( ) (D) uk () ( ) k GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

12 PAGE 6 SIGNALS & SYSEMS CHAP YEAR 4 WO MARKS MCQ.5 he rm value of he periodic waveform given in figure i (A) 6 A (B) 6 A (C) 4A / (D).5 A MCQ.5 he rm value of he reulan curren in a wire which carrie a dc curren of A and a inuoidal alernaing curren of peak value i (A) 4. A (B) 7. A (C).4 A (D). A YEAR ONE MARK MCQ.5 Fourier Serie for he waveform, f () hown in Figure i (A) 8 in( ) in( ) in( 5 ) B (B) 8 in( ) co( ) in(5 ) B (C) 8 co( ) co( ) co( 5 ) B (D) 8 co( ) in( ) in( 5 ) B MCQ.5 Le () be he ep repone of a linear yem wih zero iniial condiion; hen he repone of hi yem o an an inpu u () i (A) ( τ) u( τ) dτ (B) d ( τ) u( τ) dτ d ; E GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

13 CHAP SIGNALS & SYSEMS PAGE 7 (D) [ ( )] ( ) (C) ( τ) ; u( τ) dτedτ τ uτ dτ MCQ.54 Le Y () be he Laplace ranformaion of he funcion y, () hen he final value of he funcion i (A) LimY() (B) LimY() " " (C) Lim Y() " (D) Lim Y() " MCQ.55 Wha i he rm value of he volage waveform hown in Figure? (A) ( / ) V (B) ( / ) V (C) V (D) V YEAR ONE MARK MCQ.56 Given he relaionhip beween he inpu u () and he oupu y () o be ( τ) y () ( + τ) e u( τ) dτ, he ranfer funcion Y ()/ U () i (A) e + (C) Common daa Queion Q.5758* (B) + ( + ) (D) + 7 ( + ) Conider he volage waveform v a hown in figure GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

14 PAGE 8 SIGNALS & SYSEMS CHAP MCQ.57 MCQ.58 he DC componen of v i (A).4 (B). (C).8 (D). he ampliude of fundamenal componen of v i (A). V (B).4 V (C) V (D) V *********** GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

15 CHAP SIGNALS & SYSEMS PAGE 9 SOLUION SOL. SOL. Opion (C) i correc. n n xn [ ] b un [ ] l b l n un [ ] u[ n ] b + n n un ( ) l b l b l aking z ranform n n X6@ z n z u[ n] n / + z b u[ n] l / b l n n n n n n n / b z u[ n] n l z n + z / b z l / b l / b l n n n m z z n b l + b l b z l n m n I II III n n / / / aking m n Serie I converge if < or z z Serie II converge if z > < or z < Serie III converge if z < or z Region of convergence of () So, ROC : < z < > Xz will be inerecion of above hree Opion (D) i correc. Uing domain differeniaion propery of Laplace ranform. L If f () F () n f() So, L [ f( )] L df() d d d ; + + E + ( + + ) SOL. Opion (A) i correc. Convoluion um i defined a yn [ ] hn [ ]* gn [ ] / hngn [ ] [ k] / k For caual equence, yn [ ] hngn [ ] [ k] k yn [ ] hngn [ ] [ ] + hngn [ ] [ ] + hngn [ ] [ ] +... GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

16 PAGE SIGNALS & SYSEMS CHAP For n, y[ ] h[ ] g[ ] + h[ ] g[ ] +... h[ ] g[ ] g[ ] g[ ]... h[ ] g[ ]...(i) For n, y[ ] h[ ] g[ ] + h[ ] g[ ] + h[ ] g[ ] +... h[ ] g[ ] + h[ ] g[ ] g[ ] + g[ ] h[] b l g[ ] + g[ ] g[ ] g[ ] y[ ] From equaion (i), g[ ] h[ ] So, g[ ] SOL.4 Opion (C) i correc. ( co ω)( in ω) Hjω ( ) in in ω ω ω + ω ω We know ha invere Fourier ranform of in c funcion i a recangular funcion. So, invere Fourier ranform of Hjω ( ) h () h () + h () h() h () + h () + SOL.5 Opion (D) i correc. y () x() τ co( τ) dτ GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

17 CHAP SIGNALS & SYSEMS PAGE ime invariance : Le, x () δ() y () δ () co( τ ) d τ u () co() u () For a delayed inpu ( ) oupu i y (, ) δ( ) co( τ) dτ u () co( ) Delayed oupu y ( ) u ( ) y (, )! y ( ) Syem i no ime invarian. Sabiliy : Conider a bounded inpu x () co y () co co 6 A ", y ()" (unbounded) Syem i no able. d co 6 d SOL.6 Opion (D) i correc. f () a + ( a co ω+ b in nω) / n n he given funcion f () i an even funcion, herefore bn f () i a non zero average value funcion, o i will have a nonzero value of a a / fd () (average value of f) () ^ / h a n i zero for all even value of n and non zero for odd n a n f () co( nd ω ) ( ω) So, Fourier expanion of f () will have a and a n, n 5f,, n SOL.7 Opion (A) i correc. x () e Laplace ranformaion GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

18 PAGE SIGNALS & SYSEMS CHAP X () + y () e Y () + Convoluion in ime domain i equivalen o muliplicaion in frequency domain. z () x ()) y () Z () XY () () b + lb + l By parial fracion and aking invere Laplace ranformaion, we ge Z () + + z () e e SOL.8 Opion (D) i correc. f () L F () L f ( τ) e τ F () F () G () ) F() F () F () e E e τ F () F () aking invere Laplace ranform τ g () L [ e ] δ( τ) τ ) e F() F () F () ) " a F() F () F() SOL.9 Opion (C) i correc. h () e + e Laplace ranform of h () i.e. he ranfer funcion H () For uni ep inpu r () μ() or R () Oupu, Y () RH () () : D By parial fracion Y () + b + l GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

19 CHAP SIGNALS & SYSEMS PAGE aking invere Laplace e u() y () u () e u () u () 5. e 6 5. SOL. Opion (C) i correc. Syem i given a H () Sep inpu R () ( + ) Oupu Y () HR () () ( + ) b l aking invere Laplace ranform y () ( e ) u( ) Final value of y, () y () lim y () " ( + ) Le ime aken for ep repone o reach 98% of i final value i. So, e. 98. e ln 5 9. ec. SOL. Opion (D) i correc. Period of x, () 5. ec ω 8. SOL. Opion (B) i correc. Inpu oupu relaionhip 5 y () x() τ dτ, > Caualiy : y () depend on x( 5), > yem i noncaual. For example y() depend on x( ) (fuure value of inpu) Lineariy : Oupu i inegraion of inpu which i a linear funcion, o yem i linear. GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

20 PAGE 4 SIGNALS & SYSEMS CHAP SOL. Opion (A) i correc. Fourier erie of given funcion So, A x () A + a co nω + b in nω a x () x () an b n / n odd funcion x ()innωd / n n () in n d ( ) in n d ω + ω G / co n / co n ω ω c nω m c nω m G / ( co n) + ( co nco n) nω 6 ( ) n 4, b n n odd * n, n even So only odd harmonic will be preen in x () For econd harmonic componen ( n ) ampliude i zero. SOL.4 SOL.5 Opion (D) i correc. By parval heorem X( ) ω dω x () d X( ω) dω 4 Opion (C) i correc. Given equence xn [ ] {, }, n yn [ ] {,,,, }, n 4 If impule repone i hn [ ] hen yn [ ] hn [ ]* xn [ ] Lengh of convoluion ( yn [ ]) i o 4, xn [ ] i of lengh o o lengh of hn [ ] will be o. Le hn [ ] { abcd,,, } GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

21 CHAP SIGNALS & SYSEMS PAGE 5 Convoluion yn [ ] By comparing a { a, a+ b, b+ c, c+ d, d} a+ b & b a b+ c & c b c+ d & d c So, hn [ ] {,,,} SOL.6 Opion (D) i correc. We can oberve ha if we cale f () by a facor of and hen hif, we will ge g. () Fir cale f () by a facor of g () f ( /) Shif g () by, g () g ( ) f ` j g () f ` j SOL.7 Opion (C) i correc. g () can be expreed a GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

22 PAGE 6 SIGNALS & SYSEMS CHAP g () u ( ) u ( 5) By hifing propery we can wrie Laplace ranform of g () G () e e 5 e ( e ) SOL.8 Opion (D) i correc. Le x () L X () y () L Y () h () L H () So oupu of he yem i given a Y () XH () () L Now for inpu x ( τ) e X() (hifing propery) L τ τ h ( τ) e H() τ τ So now oupu i Y'( ) e X() $ e H() e τ y'( ) y ( τ) X() H() e τ Y() SOL.9 Opion (B) i correc. Le hree LI yem having repone H(), z H () z and H () z are Cacaded a howing below Aume H () z z + z + (noncaual) H () z z + z + (noncaual) Overall repone of he yem Hz () H () z H () z H () z Hz () ( z + z + )( z + z + ) H( z) o make Hz () caual we have o ake H () z alo caual. 6 4 Le H () z z + z + Hz () 6 4 ( z + z + )( z + z + )( z + z + ) " caual Similarly o make Hz () unable alea one of he yem hould be unable. SOL. Opion (C) i correc. Given ignal x () / k k ae jk/ GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

23 CHAP SIGNALS & SYSEMS PAGE 7 Le ω i he fundamenal frequency of ignal x () x () / jkω ae k k jω jω jω jω x () a e + a e + a + a e + a e Im[ x ( )] jω jω ( je ) + (.5 +. je ) + j+ jω jω jω jω 6e + + j6e + a ω jω jω + (.5.) e + ( + j) e j ω j ω + j ω j ω.56e + + j (co ω ) + j(jin ω ) +.5( co ω ) 64co ω in ω + co ω + 4. in + j (conan). j( jin ω ) + j SOL. Opion (A) i correc. Zranform of xn [ ] i Xz () 4z + z + 6z + z ranfer funcion of he yem Hz () z Oupu Yz () HzXz () () Yz () (z )(4z + z + 6z + z ) 4 z + 9z + 6z 8z+ 6z 8z 6z 4 + z 4z 4 z 8z + 9z 4 8z+ 8z 4z Or equence yn [ ] i yn [ ] δ[ n4] 8δ[ n ] + 9δ[ n] 4δ[ n] yn [ ] Y, n < So yn [ ] i noncaual wih finie uppor. 8 δ[ n+ ] + 8 δ[ n+ ] 4 δ[ n+ ] SOL. Opion (D) i correc. Since he given yem i LI, So principal of Superpoiion hold due o lineariy. For caual yem h (), < Boh aemen are correc. SOL. Opion (C) i correc. For an LI yem oupu i a conan muliplicaive of inpu wih ame frequency. GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

24 PAGE 8 SIGNALS & SYSEMS CHAP α Here inpu g () e in( ω) oupu y () Ke in( v+ φ) α Oupu will be in form of Ke in( ω+ φ) So \ β, v ω β SOL.4 Opion (D) i correc. Inpuoupu relaion y () x() τ d τ Caualiy : Since y () depend on x( ), So i i noncaual. imevariance : y () x( τ τ) dτ Y y( τ ) So hi i imevarian. Sabiliy : Oupu y () i unbounded for an bounded inpu. For example Le x() τ e τ ( bounded) y () τ $ τ e d τ e 8 B Unbounded SOL.5 Opion (A) i correc. Oupu y () of he given yem i y () x ()) h () Or Yjω ( ) Xj ( ω) Hj ( ω) Given ha, x () inc( α) and h () inc( β) Fourier ranform of x () and h () are Xjω ( ) F[ x ( )] rec ω, < < α ` α ω α α j Hjω ( ) F[ h ( )] rec ω, < < β ` β ω β β j Yjω ( ) rec ω rec ω αβ `α j `β j GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

25 CHAP SIGNALS & SYSEMS PAGE 9 So, Yjω ( ) K rec ω `γ j Where γ min( αβ, ) And y () Kinc( γ) SOL.6 Opion (B) i correc. Le a k i he Fourier erie coefficien of ignal x () Given y () x ( ) + x ( + ) Fourier erie coefficien of y () jkω b k e a + e a b k a cokω k k jkω b k (for all odd k ) kω, k " odd k For k, 4 k SOL.7 Opion ( ) i correc. SOL.8 SOL.9 Opion (D) i correc. Given ha Xz () z, z > a ( z a) Reidue of Xzz () n a z a i d ( z a ) X ( z ) z n z a dz d ( z a ) z z n dz ( z a ) z a d z n nz n z a dz nan Opion (C) i correc. Given ignal x () So, x () Similarly z a rec ` j, or *, elewhere x( ) rec ` j GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

26 PAGE 4 SIGNALS & SYSEMS CHAP x( ) *, or, elewhere jω jω jω jω F [ x ( ) + x( )] xe () d+ x( e ) d () e d+ () e d jω jω e e ; jω E + ; jω E j ( e ) j ( e ) jω ω + ω jω jω/ jω/ e jω/ j / ( e e ) e j / j / ( e e ) jω ω + ω jω ω jω/ jω/ jω/ jω/ ( e e )( e + e ) jω in ω $ co ω ω `j `j co ω inc ω ` j SOL. Opion (B) i correc. In opion (A) z [ n] xn [ ] z [ n] z[4 n] x[4n] yn [ ] z[ n] x[ 4n ] Y x[4 n] In opion (B) z [ n] xn [ + ] z [ n] z[4 n] x[4n+ ] yn [ ] z[ n] x[ 4n+ ] In opion (C) v [ n] x[ 4n] v [ n] v[ n] x[ 4 n] yn [ ] v[ n+ ] x[ 4( n+ )] Y x[ 4 n] In opion (D) v [ n] x[ 4n] v [ n] v[ n] x[ 4 n] yn [ ] v [ n ] x[ 4( n )] Y x[ 4 n] SOL. Opion ( ) i correc. he pecrum of ampled ignal jω ( ) conain replica of Ujω ( ) a frequencie! nf. Where n,,... f khz m ec GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

27 CHAP SIGNALS & SYSEMS PAGE 4 SOL. Opion (D) i correc. For an LI yem inpu and oupu have idenical wave hape (i.e. frequency of inpuoupu i ame) wihin a muliplicaive conan (i.e. Ampliude repone i conan) So F mu be a ine or coine wave wih ω ω SOL. Opion (C) i correc. Given ignal ha he following waveform GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

28 PAGE 4 SIGNALS & SYSEMS CHAP Funcion x() i periodic wih period and given ha x () x ( + ) (Halfwave ymmeric) So we can obain he fourier erie repreenaion of given funcion. SOL.4 Opion (C) i correc. Oupu i aid o be diorion le if he inpu and oupu have idenical wave hape wihin a muliplicaive conan. A delayed oupu ha reain inpu waveform i alo conidered diorion le. hu for diorion le oupu, inpuoupu relaionhip i given a y () Kg( ) aking Fourier ranform. Y( ω ) KG( ω) H ( ω ) & ranfer funcion of he yem So, H( ω ) Ke j ω d Ampliude repone H( ω ) K d e j ω d G( ω) H( ω) Phae repone, θn ( ω ) ω d For diorion le oupu, phae repone hould be proporional o frequency. SOL.5 Opion (A) i correc. j j Gz () z e αe ω + βe ω j ω for linear phae characeriic α β. SOL.6 Opion (A) i correc. Syem repone i given a Gz () Hz () KG() z gn [ ] δ[ n ] + δ[ n] Gz () z + z ( z + z ) So Hz () z + Kz ( + z) z Kz K For yem o be able pole hould lie inide uni circle. z z K! K + 4K K! K + 4K K + 4K K K + 4K 4 4K+ K 8K 4 K / GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

29 CHAP SIGNALS & SYSEMS PAGE 4 SOL.7 Opion (C) i correc. Given Convoluion i, h () u ( + ) ) r ( ) aking Laplace ranform on boh ide, H () L[ h ( )] L[ u ( + )] ) L[ r ( )] We know ha, L [ u ( )] / L [ u ( + )] e c m (imehifing propery) and L [ r ( )] / L r ( ) So H () H () e c m e e ; ` je; c me e c m aking invere Laplace ranform h () ( ) u( ) (imehifing propery) SOL.8 Opion (C) i correc. Impule repone of given LI yem. hn [ ] xn [ ]) yn [ ] aking z ranform on boh ide. Hz () z X() z Y() z a xn [ ] Z z xz ( ) We have Xz () z and Yz () + z So Hz () z ( z )( + z ) Oupu of he yem for inpu un [ ] δ[ n ] i, yz () HzUz () () Un [ ] Z Uz ( ) z So Yz () z ( z )( + z ) z 4 5 z ( z + z 6z ) z z + z 6z aking invere zranform on boh ide we have oupu. yn [ ] δ[ n] δ[ n ] + δ[ n4] 6δ[ n5] SOL.9 Opion (B) i correc. A bounded ignal alway poee ome finie energy. E g () d< GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

30 PAGE 44 SIGNALS & SYSEMS CHAP SOL.4 Opion (C) i correc. rigonomeric Fourier erie i given a x () A+ / an co nω+ bn in nω n Since here are no ine erm, o bn b n x ()innωd / x() τ in nωτ dτ+ x() in nωd G Where τ & dτ d / / / O / / / / / / x ( ) in nω( )( d) + x ( ) in nωd ; E O x ( ) in n d x( ) in n d ; ` ++ ω j E x ( ) in( n nω) d+ x ( ) in nωd ; E x ( ) in( nωd ) ++ x ( ) in nωd ; E b n if x () x ( ) From half wave ymmery we know ha if x () x! ` j hen Fourier erie of x () conain only odd harmonic. SOL.4 Opion (C) i correc. Z ranform of a dicree all pa yem i given a Hz () z z ) zz I ha a pole a z and a zero a /z ). Given yem ha a pole a z + % ( + j) ( +j) GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

31 CHAP SIGNALS & SYSEMS PAGE 45 yem i able if z < and for hi i i anicaual. SOL.4 Opion (A) i correc. According o given daa inpu and oupu Sequence are xn [ ] {, }, n yn [ ] {,,, }, n If impule repone of yem i hn [ ] hen oupu yn [ ] hn [ ]) xn [ ] Since lengh of convoluion ( yn [ ]) i o, xn [ ] i of lengh o o lengh of hn [ ] i o. Le hn [ ] { abc,, } Convoluion So, a yn [ ] yn [ ] { a, ab, bc, c} {,,, } a b & b a c & c Impule repone hn [ ] ",,, SOL.4 Opion ( ) i correc. SOL.44 Opion (D) i correc. Oupu y () e x () If x () i unbounded, x ()" e x () " y () ( bounded) So y () i bounded even when x () i no bounded. SOL.45 Opion (B) i correc. ' ' Given y () xd () GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

32 PAGE 46 SIGNALS & SYSEMS CHAP Laplace ranform of y () X Y () (), ha a ingulariy a ' ' For a caual bounded inpu, y () xd () i alway bounded. SOL.46 Opion (A) i correc. RMS value i given by V rm V () d Where, V () * ` j, < / So V () d d () d ` j + G / / / 4 $ d 4 ; E V rm V 6 SOL.47 SOL.48 Opion (A) i correc. By final value heorem lim f () " Opion (D) i correc. fx () ( ) lim F() lim " " ( + + ) 6 in x co x cox / fx () A+ an co nωx+ bn in nω x n fx () in xi an even funcion o b A 5. 5., n a n ), oherwie ω GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN: n

33 CHAP SIGNALS & SYSEMS PAGE 47 SOL.49 Opion (B) i correc. Zranform Fz () o, fk () δ() k ( ) hu ( ) k Z + z z z + z + + z k SOL.5 Opion (A) i correc. Roo mean quare value i given a I rm I () d From he graph, I () So, < ` j * 6, / < / Id d () 6 d ` j + G / / 44 6 e ; E + / o 44 ; 6 c + 4 m ` je [6 8 ] + 4 I rm 4 6 A SOL.5 SOL.5 Opion (B) i correc. oal curren in wire I + in ω ( ) I rm ( ) + 7. A Opion (C) i correc. Fourier erie repreenaion i given a f () A + a co nω + b in nω GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN: / n n n From he wave form we can wrie fundamenal period ec Z 4, ] ` j f () [ 4, ] ` j \ f () f( ), f () i an even funcion So, b n A fd ()

34 PAGE 48 SIGNALS & SYSEMS CHAP a n 4 d 4 ` j + ` j d G / / / 4 4 e ; E ; E / 4 4 ; c 8 m c 8 me f ()conω d 4 co co n 4 ` j ω + ` j n ω d G / By olving he inegraion 8, a n * n n i odd So,, n i even f () 8 co co( ) co( 5 ) B o / SOL.5 SOL.54 Opion (A) i correc. Repone for any inpu u () i given a y () u ()) h () h ()" impule repone y () u() τ h( τ) dτ Impule repone hand () ep repone () of a yem i relaed a h () d [ ( )] d So y () u() τ d [ τ ] d τ d u() τ ( τ) dτ d d Opion (B) i correc. Final value heorem ae ha lim y () " lim Y () " SOL.55 Opion (D) i correc. V rm () V d here () V d / / () d ( ) d ( ) d + + G / / ` + + j `j `jb 4 V GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

35 CHAP SIGNALS & SYSEMS PAGE 49 V rm V 4 SOL.56 SOL.57 Opion (D) i correc. Le h () i he impule repone of yem y () u ()) h () y () u() τ h( τ) dτ ( τ) ( + τ) e u( τ) dτ So h () ( + ) e u( ), > ranfer funcion Y () H () + U () ( + ) ( + ) ( + 7) ( + ) ( + ) Opion (B) i correc. Fourier erie repreenaion i given a / n n n v () A + a co nω + b in nω period of given wave form 5 m DC componen of v i A vd () d + d 5> H SOL.58 Opion (A) i correc. Coefficien, a n v ()conω d () co nw d + ( ) co nw d 5> H in in : n n ω ω : n n ω ω D f 5 D Pu ω 5 a n in nω in 5nω+ in nω 5 5 p GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

36 PAGE 5 SIGNALS & SYSEMS CHAP in n in 5n n; b 5 l b 5 le in 6n in n n; b 5 l ^ he in 6n n b 5 l Coefficien, b n v ()innω d pu () in nw d + ( ) in nw d 5> H co co 9 n n ω ω 9 n n ω ω C f 5 C ω 5 b n co nω+ + co 5nωco nω co nω+ + co 5nω co n co 5n n; b l b 5 le co 6n n; b l E co n 6 ; b 5 n le Ampliude of fundamenal componen of v i v f a + b a in 6 b 5 l, b co 6 b 5 l v f in 6 + co 6 5 b 5 l. Vol 5 5 p *********** GAE Previou Year Solved Paper By RK Kanodia & Ahih Murolia Publihed by: NODIA and COMPANY ISBN:

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