NODIA AND COMPANY. GATE SOLVED PAPER Electrical Engineering SIGNALS & SYSTEMS. Copyright By NODIA & COMPANY

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1 No par of hi publicaion may be reproduced or diribued in any form or any mean, elecronic, mechanical, phoocopying, or oherie ihou he prior permiion of he auhor. GAE SOLVED PAPER Elecrical Engineering SIGNALS & SYSEMS Copyrigh By NODIA & COMPANY Informaion conained in hi book ha been obained by auhor, from ource believe o be reliable. Hoever, neiher Nodia nor i auhor guaranee he accuracy or compleene of any informaion herein, and Nodia nor i auhor hall be reponible for any error, omiion, or damage ariing ou of ue of hi informaion. hi book i publihed ih he underanding ha Nodia and i auhor are upplying informaion bu are no aemping o render engineering or oher profeional ervice. NODIA AND COMPANY B8, Dhanhree oer I, Cenral Spine, Vidyadhar Nagar, Jaipur 9 Ph : nodia.co.in enquiry@nodia.co.in

2 GAE SOLVED PAPER EE SIGNALS & SYSEMS YEAR ONE MARK Q. A bandlimied ignal ih a maximum frequency of 5kH i o be ampled. According o he ampling heorem, he ampling frequency hich i no valid i (A) 5kH (B) kh (C) 5 kh (D) kh Q. For a periodic ignal v ^ h in + co + 6 in^5+p/ 4h, he fundamenal frequency in rad/ (A) (B) (C) 5 (D) 5 Q. o yem ih impule repone h^h and h^h are conneced in cacade. hen he overall impule repone of he cacaded yem i given by (A) produc of h^h and h^h (B) um of h^h and h^h (C) convoluion of h^h and h^h (D) ubracion of h^h from h^h Q. 4 Which one of he folloing aemen i NO RUE for a coninuou ime caual and able LI yem? (A) All he pole of he yem mu lie on he lef ide of he j axi (B) Zero of he yem can lie anyhere in he plane (C) All he pole mu lie ihin (D) All he roo of he characeriic equaion mu be locaed on he lef ide of he j axi. Q. 5 he impule repone of a yem i h^h u ^ h. For an inpu u^ h, he oupu i (A) u ^ h (B) ^ h u ^ h (C) ^ h u ^ h (D) u ^ h YEAR WO MARKS Q. 6 he impule repone of a coninuou ime yem i given by h^h d^ h+ d^h. he value of he ep repone a i (A) (B) (C) (D)

3 GAE SOLVED PAPER EE SIGNALS & SYSEMS YEAR ONE MARK n n Q. 7 If xn [ ] (/) (/) un [ ], hen he region of convergence (ROC) of i ranform in he plane ill be (A) < < (B) < < (C) < < (D) < Q. 8 he unilaeral Laplace ranform of f () i. he unilaeral Laplace + + ranform of f() i (A) (B) + ( + + ) ( + + ) (C) ( + + ) YEAR (D) + ( + + ) WO MARKS Q. 9 Le yn [ ] denoe he convoluion of hn [ ] and gn, [ ] here hn [ ] ( / ) un [ ] and gn [ ] i a caual equence. If y [] and y [] /, hen g [] equal (A) (B) / (C) (D) / Q. he Fourier ranform of a ignal h () i Hj ( ) ( co )( in )/. he value of h() i (A) 4 / (B) / (C) (D) Q. he inpu x () and oupu y () of a yem are relaed a y () x( ) co( ) d. he yem i (A) imeinvarian and able (B) able and no imeinvarian (C) imeinvarian and no able (D) no imeinvarian and no able YEAR ONE MARK Q. he Fourier erie expanion f () a + / an con + bninn of n he periodic ignal hon belo ill conain he folloing nonero 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

4 GAE SOLVED PAPER EE SIGNALS & SYSEMS Q. Given o coninuou ime ignal x () e and y () e hich exi for > *, he convoluion () x () y () i (A) e e (B) e (C) e + (D) e + e YEAR WO MARKS Q. 4 Le he Laplace ranform of a funcion f () hich exi for > be F () and he Laplace ranform of i delayed verion f ( ) be F (). Le F *( ) be he complex conjugae of F () ih he Laplace variable e + j. If F() F*() G (), hen he invere Laplace ranform of G () i an ideal F () (A) impule d () (B) delayed impule d( ) (C) ep funcion u () (D) delayed ep funcion u ( ) Q. 5 he repone h () of a linear ime invarian yem o an impule d (), 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() YEAR (D) e d () + e u() ONE MARK Q. 6 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 Q. 7 he period of he ignal x ( ) 8 in.8p p ` + 4 j i (A).4p (B).8p (C).5 (D).5 Q. 8 he yem repreened by he inpuoupu relaionhip y () x( ) d, > i (A) Linear and caual (B) Linear bu no caual (C) Caual bu no linear (D) Neiher liner nor caual 5 Q. 9 he econd harmonic componen of he periodic aveform given in he figure ha an ampliude of (A) (B) (C) / p (D) 5

5 GAE SOLVED PAPER EE SIGNALS & SYSEMS Q. () YEAR WO MARKS x i a poiive recangular pule from o + ih uni heigh a hon in he figure. he value of X( ) d " here X( ) i he Fourier ranform of x ()} i. (A) (C) 4 (B) p (D) 4p Q. Given he finie lengh inpu xn [ ] and he correponding finie lengh oupu yn [ ] of an LI yem a hon belo, he impule repone hn [ ] of he yem i (A) hn [ ] {,,, } (B) hn [ ] {,, } (C) hn [ ] {,,, } (D) hn [ ] {,, } Common Daa Queion Q.. Given f () and g () a ho belo Q. g () can be expreed a (A) g () f( ) (B) g () f ` j (C) g () f ` j (D) g () f ` j Q. 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 Q. 4 A Linear ime Invarian yem ih an impule repone h () produce oupu y () hen inpu x () i applied. When he inpu x ( ) i applied o a yem ih impule repone h ( ), he oupu ill be (A) y() (B) y( ( )) (C) y ( ) (D) y ( )

6 GAE SOLVED PAPER EE SIGNALS & SYSEMS YEAR 9 WO MARKS Q. 5 A cacade of hree Linear ime Invarian yem i caual and unable. From hi, e 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 (D) he majoriy are unable and he majoriy are caual Q. 6 he Fourier Serie coefficien of a periodic ignal x () expreed a jpk/ 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 folloing i rue? (A) x () ha finie energy becaue only finiely many coefficien are nonero (B) x () ha ero average value becaue i i periodic (C) he imaginary par of x () i conan (D) he real par of x () i even Q. 7 he ranform of a ignal xn [ ] i given by I i applied o a yem, ih a ranfer funcion H () Le he oupu be yn. [ ] Which of he folloing i rue? (A) yn [ ] i non caual ih finie uppor (B) yn [ ] i caual ih infinie uppor (C) yn [ ] ; n > (D) Re[ Y ( )] ji e Re[ Y ( )] ji e Im[ Y ( )] ji Im[ Y ( )] j i; p q < p YEAR 8 e e ONE MARK Q. 8 he impule repone of a caual linear imeinvarian yem i given a h. () No conider he folloing o aemen : Saemen (I): Principle of uperpoiion hold Saemen (II): h () for < Which one of he folloing aemen i correc? (A) Saemen (I) i correc and aemen (II) i rong (B) Saemen (II) i correc and aemen (I) i rong (C) Boh Saemen (I) and Saemen (II) are rong (D) Boh Saemen (I) and Saemen (II) are correc a Q. 9 A ignal e in( ) i he inpu o a real Linear ime Invarian yem. Given K b and f are conan, he oupu of he yem ill be of he form Ke in( v + f) here (A) b need no be equal o a bu v equal o (B) v need no be equal o bu b equal o a (C) b equal o a and v equal o (D) b need no be equal o a and v need no be equal o

7 GAE SOLVED PAPER EE SIGNALS & SYSEMS YEAR 8 WO MARKS Q. A yem ih x () and oupu y () i defined by he inpuoupu relaion : y () xd () he yem ill be (A) Caual, imeinvarian and unable (B) Caual, imeinvarian and able (C) noncaual, imeinvarian and unable (D) noncaual, imevarian and unable Q. A ignal x () inc( a) here a i a real conan ^inc() x h i he inpu o a Linear ime Invarian yem hoe impule repone h () inc( b), here b i a real conan. If min ( ab, ) denoe he minimum of a and b and imilarly, max ( ab, ) denoe he maximum of a and b, and K i a conan, hich one of he folloing aemen i rue abou he oupu of he yem? (A) I ill be of he form Kinc( g ) here g min( a, b) (B) I ill be of he form Kinc( g ) here g max( a, b) (C) I ill be of he form Kinc( a) (D) I can no be a inc ype of ignal in( x) x p p Q. Le x () be a periodic ignal ih 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) / Q. () (D) H i a ranfer funcion of a real yem. When a ignal xn [ ] ( + ) i he inpu o uch a yem, he oupu i ero. Furher, he Region of convergence (ROC) of ^ h H() i he enire Zplane (excep ). I can hen be inferred ha H () can have a minimum of (A) one pole and one ero (B) one pole and o ero (C) o pole and one ero D) o pole and o ero j n Q. 4 Given X () ih > a, he reidue of X () n a a for n $ ill be ( a) (A) a n (B) a n (C) na n (D) na n Q. 5 Le x () rec^ h (here rec() x for x and ero oherie. in( x) x If inc() x p p, hen he Fof x () + x( ) ill be given by (A) inc ` p j (B) inc ` p j (C) inc co ` p j ` j (D) inc in ` p j ` j

8 GAE SOLVED PAPER EE SIGNALS & SYSEMS Q. 6 Given a equence xn, [ ] o generae he equence yn [ ] x[ 4n], hich one of he folloing procedure ould be correc? (A) Fir delay xn ( ) by ample o generae [ n], hen pick every 4 h ample of [ n] o generae [ n], and han finally ime revere [ n] o obain yn. [ ] (B) Fir advance xn [ ] by ample o generae [ n], hen pick every 4 h ample of [ n] o generae [ n], and hen finally ime revere [ 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 Q. 7 Le a ignal ain( + f) be applied o a able linear ime varian yem. Le he correponding eady ae oupu be repreened a af ( + f). hen hich of he folloing aemen i rue? (A) F i no necearily a Sine or Coine funcion bu mu be periodic ih. (B) F mu be a Sine or Coine funcion ih a a (C) F mu be a Sine funcion ih and f f (D) F mu be a Sine or Coine funcion ih Q. 8 he frequency pecrum of a ignal i hon in he figure. If hi i ideally ampled a inerval of m, hen he frequency pecrum of he ampled ignal ill be

9 GAE SOLVED PAPER EE SIGNALS & SYSEMS YEAR 7 WO MARKS Q. 9 A ignal x () i given by, / 4 < / 4 x () *, / 4< 7/ 4 x ( + ) Which among he folloing give he fundamenal fourier erm of x ()? (A) 4 co p p p ` 4 j (B) p co 4 ` p + p 4j (C) 4 in p p p ` 4 j (D) p in 4 ` p + p 4j Common Daa Queion Q. 4 4 A ignal i proceed by a caual filer ih ranfer funcion G () Q. 4 For a diorion free oupu ignal ave form, G () mu (A) provide ero phae hif for all frequency (B) provide conan phae hif for all frequency (C) provide linear phae hif ha i proporional o frequency (D) provide a phae hif ha i inverely proporional o frequency

10 GAE SOLVED PAPER EE SIGNALS & SYSEMS Q. 4 () G a + b i a lo pa digial filer ih a phae characeriic ame a ha of he above queion if (A) a b (B) a b ( / ) (C) a b (D) a ( / ) b Q. 4 Conider he dicreeime yem hon in he figure here he impule repone of G () i g(), g() g(), g() g(4) g hi yem i able for range of value of K (A) [, ] (B) [, ] (C) [, ] (D) [, ] Q. 4 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 Q. 44 X (), Y () + are Z ranform of o ignal xn [ ], yn [ ] repecively. A linear ime invarian yem ha he impule repone hn [ ] defined by hee o ignal a hn [ ] xn [ ]* yn [ ] here * denoe dicree ime convoluion. hen he oupu of he yem for he inpu d[ n ] (A) ha Zranform X() Y() (B) equal d[ n] d[ n ] + d[ n4] 6d[ n5] (C) ha Zranform + 6 (D) doe no aify any of he above hree YEAR 6 ONE MARK Q. 45 he folloing i rue (A) A finie ignal i alay bounded (B) A bounded ignal alay poee finie energy (C) A bounded ignal i alay ero ouide he inerval [, ] for ome (D) A bounded ignal i alay finie Q. 46 () x i a real valued funcion of a real variable ih period. I rigonomeric Fourier Serie expanion conain no erm of frequency p( 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 ( / ) Q. 47 A dicree real all pa yem ha a pole a + % : i, herefore (A) alo ha a pole a + % (B) ha a conan phae repone over he plane: arg H () conan

11 GAE SOLVED PAPER EE SIGNALS & SYSEMS conan (C) i able only if i i anicaual (D) ha a conan phae repone over he uni circle: arg He ( ) iw conan YEAR 6 WO MARKS Q. 48 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 [ ] ill be (A) hn [ ] ; 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[ ] Q. 49 he dicreeime ignal xn [ ] X ( ) n n n /, here denoe a n + ranformpair relaionhip, i orhogonal o he ignal / / / n (A) y[ n] ) Y( ) n `j n n (B) y [ n] ) Y ( ) ( 5 n) (C) y [ n] ) Y ( ) n n n n (D) y [ n] ) Y ( ) ( n + ) Q. 5 A coninuouime yem i decribed by y () e x (), here y () i he oupu and x () i he inpu. y () i bounded (A) only hen x () i bounded (B) only hen x () i nonnegaive (C) only for if x () i bounded for $ (D) even hen x () i no bounded Q. 5 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 eroe in i double ided Laplace ranform Y () ' ' YEAR 5 WO MARKS Q. 5 For he riangular ave from hon in he figure, he RMS value of he volage i equal o

12 GAE SOLVED PAPER EE SIGNALS & SYSEMS (A) (C) 6 (B) (D) Q. 5 he Laplace ranform of a funcion f () i F () 5 6 ", approache ( + + ) f ( ) (A) (B) 5 (C) 7 (D) Q. 54 he Fourier erie for he funcion fx () inxi (A) in x+ in x (B) co x (C) in x+ co x (D) cox Q. 55 If u () i he uni ep and d () i he uni impule funcion, he invere ranform of F () + for k > i (A) ( ) k d( k) (B) d() k () k (C) ( ) k uk ( ) (D) uk ()( ) YEAR 4 k WO MARKS Q. 56 he rm value of he reulan curren in a ire hich 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 Q. 57 he rm value of he periodic aveform given in figure i (A) 6 A (B) 6 A (C) 4A / (D).5 A ***********

13 GAE SOLVED PAPER EE SIGNALS & SYSEMS SOLUION Sol. Sol. Sol. Opion (A) i correc. Given, he maximum frequency of he bandlimied ignal 5kH f m According o he Nyqui ampling heorem, he ampling frequency mu be greaer han he Nyqui frequency hich i given a f N f m 5 kh So, he ampling frequency f mu aify f $ f N $ kh f only he opion (A) doe no aify he condiion herefore, 5kH i no a valid ampling frequency. Opion (A) i correc. Given, he ignal v ^ h in + co + 6 in^5+ p 4 h So e have rad/ ; rad/ and 5 rad/ herefore, he repecive ime period are ec p p ec p p p ec 5 So, he fundamenal ime period of he ignal i LCM^ p, p,ph L.C.M. ^, h HCF^,, 5h or, p hu, he fundamenal frequency in rad/ec i p rad/ Opion (C) i correc. If he o yem ih impule repone h^h and h^h are conneced in cacaded configuraion a hon in figure, hen he overall repone of he yem i he convoluion of he individual impule repone.

14 GAE SOLVED PAPER EE SIGNALS & SYSEMS Sol. 4 Sol. 5 Sol. 6 Opion (C) i correc. For a yem o be caual, he R.O.C of yem ranfer funcion H^h hich i raional hould be in he righ half plane and o he righ of he righ mo pole. For he abiliy of LI yem. All pole of he yem hould lie in he lef half of S plane and no repeaed pole hould be on imaginary axi. Hence, opion (A), (B), (D) aifie boh abiliy and caualiy an LI yem. Bu, Opion (C) i no rue for he able yem a, S have one pole in righ hand plane alo. Opion (C) i correc. Given, he inpu x ^ h u ^ h I Laplace ranform i X ^ h e he impule repone of yem i given h ^ h u^h I Laplace ranform i H ^ h Hence, he overall repone a he oupu i e Y ^ h XH ^ h ^ h I invere Laplace ranform i ^ y ^ h h u ^ h Opion (B) i correc. Given, he impule repone of coninuou ime yem h ^ h d^ h+ d^h From he convoluion propery, e kno x ^ h* d^ h x ^ h So, for he inpu x ^ h u ^ h (Uni ep fun n ) he oupu of he yem i obained a y ^ h u ^ h* h ^ h Sol. 7 u ^ h* 6 d^ h+ d^h@ u ^ h+ u ^ h A y^h u^ h+ u^h 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 ranform n n X6@ n u[ n] n / + b u[ n] l / b l n n

15 GAE SOLVED PAPER EE SIGNALS & SYSEMS Sol. 8 Sol. 9 / n n b u[ n] l n n n n n n + / b l / b l / b l n n Serie I converge if < or Serie II converge if n n n m / / / aking m n n b l + b l b l n m n I II III > < or < Serie III converge if < or Region of convergence of () So, ROC : < < > X ill be inerecion of above hree Opion (D) i correc. Uing domain differeniaion propery of Laplace ranform. If f () F () L df() f() d So, L [ f( )] d d ; + + E + ( + + Opion (A) i correc. ) Convoluion um i defined a yn [ ] hn [ ]* gn [ ] / hngn [ ] [ k] For caual equence, yn [ ] hngn [ ] [ k] L / k k yn [ ] hngn [ ] [ ] + hngn [ ] [ ] + hngn [ ] [ ] +... 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[ ] Sol. g[ ] + g[ ] h[] b l g[ ] + g[ ] g[ ] g[ ] y[ ] From equaion (i), g[ ] h[ ] So, g[ ] Opion (C) i correc. ( co )( in ) Hj ( ) in in + We kno ha invere Fourier ranform of in c funcion i a recangular funcion.

16 GAE SOLVED PAPER EE SIGNALS & SYSEMS Sol. Sol. So, invere Fourier ranform of Hj ( ) Opion (D) i correc. y () x() co( ) d ime invariance : Le, x () d() h () h() + h() h() h() + h() + y () d () co( ) d u () co() u () For a delayed inpu ( ) oupu i Delayed oupu y (, ) d( ) co( ) d u () co( ) 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. Opion (D) i correc. d co 6 d f () a + ( a co + b in n) n he given funcion f () i an even funcion, herefore b / n n n

17 GAE SOLVED PAPER EE SIGNALS & SYSEMS Sol. Sol. 4 Sol. 5 f () i a non ero average value funcion, o i ill have a nonero value of a / a fd () (average value of f) () ^ / h a n i ero for all even value of n and non ero for odd n a n f () co( nd ) ( ) So, Fourier expanion of f () ill have a and a n, n 5f,, Opion (A) i correc. x () e Laplace ranformaion X () + y () e Y () + Convoluion in ime domain i equivalen o muliplicaion in frequency domain. () x ()) y () Z () XY () () b + lb + l By parial fracion and aking invere Laplace ranformaion, e ge Z () + + Opion (D) i correc. () e e L f () F () L f ( ) e F () F () ) F() F () G () F () e E e F () F () aking invere Laplace ranform g () L [ e ] d( ) Opion (C) i correc. h () e + e ) e F() F () F () ) " a F() F () F() Laplace ranform of h () i.e. he ranfer funcion H () For uni ep inpu r () m() or R () Oupu, Y () RH () () : D By parial fracion

18 GAE SOLVED PAPER EE SIGNALS & SYSEMS Sol. 6 Sol. 7 Sol. 8 Sol. 9 Y () + b + l aking invere Laplace e u() y () u () e u () Opion (C) i correc. Syem i given a H () Sep inpu R () u () 5. e 6 5. ( + ) 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 Opion (D) i correc. Period of x, () p Opion (B) i correc. Inpu oupu relaionhip ln 5 9. ec. 5 p 5. ec 8. p y () x() d, > 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 hich i a linear funcion, o yem i linear. Opion (A) i correc. Fourier erie of given funcion So, A x () A + a co n + b in n a x () x () an b n / n odd funcion x ()innd n n

19 GAE SOLVED PAPER EE SIGNALS & SYSEMS Sol. Sol. / () in n d ( ) in n d + G / co n / co n c n m c n m G / ( co np) + ( co npco np) n 6 ( ) np 4, b n n odd * np, n even So only odd harmonic ill be preen in x () For econd harmonic componen ( n ) ampliude i ero. Opion (D) i correc. By parval heorem X( ) d x () d p X( ) d p 4p 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 [ ] ill be o. Le hn [ ] { abcd,,, } Convoluion By comparing yn [ ] So, hn [ ] { a, a+ b, b+ c, c+ d, d} a a+ b & b a b+ c & c b c+ d & d c {,,,}

20 GAE SOLVED PAPER EE SIGNALS & SYSEMS Sol. Opion (D) i correc. We can oberve ha if e cale f () by a facor of and hen hif, e ill ge g. () Fir cale f () by a facor of g () f ( /) Shif g () by, g () g ( ) f ` j Sol. Sol. 4 Sol. 5 g () f ` j Opion (C) i correc. g () can be expreed a g () u ( )u ( 5) By hifing propery e can rie Laplace ranform of g () G () e e 5 Opion (D) i correc. Le x () L X () y () L Y () h () L H () So oupu of he yem i given a e ( e ) Y () XH () () No for inpu x ( ) e X() (hifing propery) L L h ( ) e H() So no oupu i Y'( ) e X() $ e H() e X() H() e Y() y'( ) y ( ) Opion (B) i correc. Le hree LI yem having repone H(), H () and H () are Cacaded a hoing belo Aume H () H () + + (noncaual) + + (noncaual)

21 GAE SOLVED PAPER EE SIGNALS & SYSEMS Overall repone of he yem H () H () H () H () H () ( + + )( + + ) H( ) o make H () caual e have o ake H () alo caual. 6 4 Le H () + + H () 6 4 ( + + )( + + )( + + ) " caual Sol. 6 Sol. 7 Sol. 8 Similarly o make H () unable alea one of he yem hould be unable. Opion (C) i correc. Given ignal x () / k k ae jpk/ Le i he fundamenal frequency of ignal x () x () / jk ae k k j j j j j j x () a e + a e + a + a e + a e ( je ) + (.5 +. je ) + j+ j j j j 6e + + j6e + a p j j + (.5.) e + ( + j) e j j + j j.56e + + j (co ) + j(jin ) +.5( co ). j( jin ) + j 64co in + co j Im[ x ( )] (conan) Opion (A) i correc. Zranform of xn [ ] i X () ranfer funcion of he yem H () Oupu Y () HX () () Y () ( )( ) Or equence yn [ ] i yn [ ] d[ n4] 8d[ n ] + 9d[ n] 4d[ n] yn [ ] Y, n < So yn [ ] i noncaual ih finie uppor. 8 d[ n+ ] + 8 d[ n+ ] 4 d[ n+ ] 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.

22 GAE SOLVED PAPER EE SIGNALS & SYSEMS Sol. 9 Sol. Sol. Opion (C) i correc. For an LI yem oupu i a conan muliplicaive of inpu ih ame frequency. Here inpu g () e a in( ) oupu y () Ke in( v+ f) a Oupu ill be in form of Ke in( + f) So \ b, v Opion (D) i correc. Inpuoupu relaion y () x() d Caualiy : Since y () depend on x( ), So i i noncaual. imevariance : y () x( ) d Y y( ) So hi i imevarian. Sabiliy : Oupu y () i unbounded for an bounded inpu. For example Le x() e ( bounded) b $ y () e d e 8 B Opion (A) i correc. Oupu y () of he given yem i y () x ()) h () Or Yj ( ) Xj ( ) Hj ( ) Unbounded Given ha, x () inc( a) and h () inc( b) Fourier ranform of x () and h () are Xj ( ) F[ x ( )] p rec, < < a ` a a a j Hj ( ) F[ h ( )] p rec, < < b ` b b b j Yj ( ) p rec rec ab `a j `b j Sol. So, Yj ( ) K rec `g j Where g min( ab, ) And y () Kinc( g) Opion (B) i correc.

23 GAE SOLVED PAPER EE SIGNALS & SYSEMS Sol. Sol. 4 Sol. 5 Le a k i he Fourier erie coefficien of ignal x () Given y () x ( ) + x ( + ) Fourier erie coefficien of y () jk b k e a + e a b k a cok k k jk b k (for all odd k ) k p, k " odd k p p For k, 4 Opion ( ) i correc. Opion (D) i correc. Given ha X (), > a ( a) Reidue of X () n a a i d ( a ) X ( ) n a d d ( a ) n d ( a ) a d n n n a d nan a Opion (C) i correc. Given ignal x () rec ` j, or So, x () *, elehere Similarly x( ) rec ` j, or x( ) *, elehere k j j j j F [ x ( ) + x( )] xe () d+ x( e ) d () e d+ () e d j j e e ; j E + ; j E j ( e ) j ( e ) j + j j/ j/ e j/ j / ( e e ) e j / j / ( e e ) j + j j/ j/ j/ j/ ( e e )( e + e ) j in $ co ` j ` j co inc `p j

24 GAE SOLVED PAPER EE SIGNALS & SYSEMS Sol. 6 Sol. 7 Opion (B) i correc. In opion (A) [ n] xn [ ] In opion (B) In opion (C) In opion (D) [ n] [4 n] x[4n] yn [ ] [ n] x[ 4n ] Y x[4 n] [ n] xn [ + ] [ n] [4 n] x[4n+ ] yn [ ] [ n] x[ 4n+ ] v [ n] x[ 4n] v [ n] v[ n] x[ 4 n] yn [ ] v [ n+ ] x[ 4( n+ )] Y x[4 n] v [ n] x[ 4n] v [ n] v[ n] x[ 4 n] yn [ ] v [ n ] x[ 4( n )] Y x[4 n] Opion ( ) i correc. he pecrum of ampled ignal j ( ) conain replica of Uj ( ) a frequencie! nf. Where n,,... f kh m ec Sol. 8 Opion (D) i correc. For an LI yem inpu and oupu have idenical ave hape (i.e. frequency of inpuoupu i ame) ihin a muliplicaive conan (i.e. Ampliude repone

25 GAE SOLVED PAPER EE SIGNALS & SYSEMS Sol. 9 i conan) So F mu be a ine or coine ave ih Opion (C) i correc. Given ignal ha he folloing aveform Sol. 4 Sol. 4 Sol. 4 Funcion x() i periodic ih period and given ha x () x ( + ) (Halfave ymmeric) So e can obain he fourier erie repreenaion of given funcion. Opion (C) i correc. Oupu i aid o be diorion le if he inpu and oupu have idenical ave hape ihin a muliplicaive conan. A delayed oupu ha reain inpu aveform i alo conidered diorion le. hu for diorion le oupu, inpuoupu relaionhip i given a y () Kg( ) aking Fourier ranform. Y( ) KG( ) e j d G( ) H( ) H ( ) & ranfer funcion of he yem So, H( ) Ke j d d Ampliude repone H( ) K Phae repone, qn ( ) d For diorion le oupu, phae repone hould be proporional o frequency. Opion (A) i correc. j j G () e ae + be j for linear phae characeriic a b. Opion (A) i correc. Syem repone i given a G () H () KG() So H () gn [ ] d[ n ] + d[ n] G () + ( + ) K ( + ) + K K

26 GAE SOLVED PAPER EE SIGNALS & SYSEMS Sol. 4 For yem o be able pole hould lie inide uni circle. K! K + 4K K! K + 4K K + 4K K K + 4K 4 4K+ K 8K 4 K / Opion (C) i correc. Given Convoluion i, h () u ( + ) ) r ( ) aking Laplace ranform on boh ide, H () L[ h ( )] L[ u ( + )] ) L[ r ( )] Sol. 44 Sol. 45 We kno ha, L [ u ( )] / L [ u ( + )] e c m (imehifing propery) and L [ r ( )] / L[ r ( ) e c m So H () e e ; ` je; c me H () e c m aking invere Laplace ranform h () ( ) u( ) (imehifing propery) Opion (C) i correc. Impule repone of given LI yem. hn [ ] xn [ ]) yn [ ] aking ranform on boh ide. H () X() Y() a xn [ ] Z x ( ) We have X () and Y () + So H () ( )( + ) Oupu of he yem for inpu un [ ] d[ n ] i, y () HU () () Un [ ] Z U ( ) So Y () ( )( + ) 4 5 ( + 6 ) + 6 aking invere ranform on boh ide e have oupu. yn [ ] d[ n] d[ n ] + d[ n4] 6d[ n5] Opion (B) i correc. A bounded ignal alay poee ome finie energy. E g () d<

27 GAE SOLVED PAPER EE SIGNALS & SYSEMS Sol. 46 Sol. 47 Opion (C) i correc. rigonomeric Fourier erie i given a x () A+ / an co n+ bn in n n Since here are no ine erm, o bn b n x ()innd / Where & d d x() in n d+ x() in nd G / / / O / / / / / / x ( ) in n( )( d) + x ( ) in nd ; E O x ( ) in n p d x( ) in n d ; ` ++ j E x ( ) in( np n) d+ x ( ) in nd ; E x ( ) in( nd ) ++ x ( ) in nd ; E b n if x () x ( ) From half ave ymmery e kno ha if x () x! ` j hen Fourier erie of x () conain only odd harmonic. Opion (C) i correc. Z ranform of a dicree all pa yem i given a H () ) I ha a pole a and a ero a / ). Given yem ha a pole a + % ( + j) ( +j) Sol. 48 yem i able if < and for hi i i anicaual. Opion (A) i correc. According o given daa inpu and oupu Sequence are xn [ ] {, }, n yn [ ] {,,,}, n

28 GAE SOLVED PAPER EE SIGNALS & SYSEMS 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 Sol. 49 Sol. 5 Sol. 5 Sol. 5 yn [ ] yn [ ] {a, ab, bc, c} {,,,} So, a a b & b a c & c Impule repone hn [ ] ",,, Opion ( ) i correc. Opion (D) i correc. Oupu y () e x () If x () i unbounded, x () " e x () " y () ( bounded) So y () i bounded even hen x () i no bounded. Opion (B) i correc. ' ' Given y () xd () Laplace ranform of y () X Y () (), ha a ingulariy a ' ' For a caual bounded inpu, y () xd () i alay bounded. 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 4 ; E 4 6

29 GAE SOLVED PAPER EE SIGNALS & SYSEMS Sol. 5 Sol. 54 Sol. 55 Sol. 56 Sol. 57 V rm V 6 Opion (A) i correc. By final value heorem ( ) lim f () lim F() lim " " ( + + ) Opion (D) i correc. fx () in x co x " cox fx () A+ / an co nx+ bn in n x n fx () in xi an even funcion o bn A 5. 5., n a n ), oherie p p Opion (B) i correc. Zranform F () o, fk () d() k () hu ( ) k Z + Opion (B) i correc. oal curren in ire I + in ( ) I rm ( ) + 7. A Opion (A) i correc. Roo mean quare value i given a I rm I () d, From he graph, I () < ` j * 6, / < So / Id d () 6 d ` + j G 44 e 6 ; E + 44 ; 6 c + 4 m ` je [6 8 ] + 4 k / I rm 4 6 A / / o ***********