UNIT 6 Signals and Systems

Transcription

1 UNIT 6 ONE MARK MCQ 6. dy dy The differeial equaio y x( ) d d + describes a sysem wih a ipu x () ad a oupu y. () The sysem, which is iiially relaxed, is excied by a ui sep ipu. The oupu y^h ca be represeed by he waveform MCQ 6. The rigoomeric Fourier series of a eve fucio does o have he (A) dc erm (B) cosie erms (C) sie erms (D) odd harmoic erms Published by: NODIA ad COMPANY ISBN:

2 MCQ 6. A sysem is defied by is impulse respose h ( ) u ( ). The sysem is (A) sable ad causal (B) causal bu o sable (C) sable bu o causal (D) usable ad o-causal MCQ 6.4 If he ui sep respose of a ework is ( e α ), he is ui impulse respose is (A) αe α (B) α α e (C) ( α α )e α (D) ( α)e TWO MARKS MCQ 6.5 A ipu x () exp( ) u () + δ( 6) is applied o a LTI sysem wih impulse respose h () u (). The oupu is (A) [ exp( )] u( ) + u( + 6) (B) [ exp( )] u( ) + u( 6) (C) 5. [ exp( )] u( ) + u( + 6) (D) 5. [ exp( )] u( ) + u( 6) MCQ 6.6 Two sysems H( Z) ad H ( Z) are coeced i cascade as show below. The overall oupu y ( ) is he same as he ipu x ( ) wih a oe ui delay. The rasfer fucio of he secod sysem H ( Z) is (A). 6z z (. 4z ) (B) z (. 6z ) (. 4z ) (C) z (. 4z ) (. 6z ) (D). 4z z (. 6z ) Page 64 Published by: NODIA ad COMPANY ISBN:

3 MCQ 6.7 The firs six pois of he 8-poi DFT of a real valued sequece are 5, j,, j4, ad + j4. The las wo pois of he DFT are respecively (A), j (B), + j (C) + j, 5 (D) j, 5 ONE MARK MCQ 6.8 The rigoomeric Fourier series for he waveform f () show below coais (A) oly cosie erms ad zero values for he dc compoes (B) oly cosie erms ad a posiive value for he dc compoes (C) oly cosie erms ad a egaive value for he dc compoes (D) oly sie erms ad a egaive value for he dc compoes MCQ 6.9 Cosider he z -rasform xz ( ) 5z + 4z + ; < z <. The iverse z - rasform x [ ] is (A) 5 δ[ + ] + δ[ ] + 4δ[ ] (B) 5 δ[ ] + δ[ ] + 4 δ[ + ] (C) 5u [ + ] + u [ ] + 4u [ ] (D) 5u [ ] + u [ ] + 4u [ + ] Published by: NODIA ad COMPANY ISBN: Page 65

4 MCQ 6. Two discree ime sysem wih impulse respose h[ ] δ[ ] ad h[ ] δ[ ] are coeced i cascade. The overall impulse respose of he cascaded sysem is (A) δ[ ] + δ[ ] (B) δ[ 4] (C) δ[ ] (D) δ[ ] δ[ ] MCQ 6. For a N -poi FET algorihm N m which oe of he followig saemes is TRUE? (A) I is o possible o cosruc a sigal flow graph wih boh ipu ad oupu i ormal order (B) The umber of buerflies i he m h sage i N/m (C) I-place compuaio requires sorage of oly N daa (D) Compuaio of a buerfly requires oly oe complex muliplicaio. TWO MARKS MCQ 6. Give f () L s + ; s 4 s + + ( k ) s E. If lim f (), he he value " of k is (A) (B) (C) (D) 4 MCQ 6. A coiuous ime LTI sysem is described by d y() dy() dx() y ( ) + 4x () d d d Page 66 Assumig zero iiial codiios, he respose y () of he above sysem for he ipu x () e u () is give by (A) ( e e ) u( ) (B) ( e e ) u( ) (C) ( e + e ) u( ) (D) ( e + e ) u( ) Published by: NODIA ad COMPANY ISBN:

5 MCQ 6.4 The rasfer fucio of a discree ime LTI sysem is give by Hz () z 4 z + z 4 8 Cosider he followig saemes: S: The sysem is sable ad causal for ROC: z > / S: The sysem is sable bu o causal for ROC: z < / 4 S: The sysem is eiher sable or causal for ROC: 4 / < z < / Which oe of he followig saemes is valid? (A) Boh S ad S are rue (B) Boh S ad S are rue (C) Boh S ad S are rue (D) S, S ad S are all rue 9 ONE MARK MCQ 6.5 The Fourier series of a real periodic fucio has oly (P) cosie erms if i is eve (Q) sie erms if i is eve (R) cosie erms if i is odd (S) sie erms if i is odd Which of he above saemes are correc? (A) P ad S (B) P ad R (C) Q ad S (D) Q ad R MCQ 6.6 A fucio is give by f ( ) si + cos. Which of he followig is rue? (A) f has frequecy compoes a ad π Hz (B) f has frequecy compoes a ad π Hz (C) f has frequecy compoes a π ad π Hz (D) f has frequecy compoes a. π ad π Hz Published by: NODIA ad COMPANY ISBN: Page 67

6 MCQ 6.7 The ROC of z -rasform of he discree ime sequece x ( ) u ( ) b u( ) l b l is (A) < z < (B) z > (C) z < (D) < z < 9 TWO MARKS MCQ 6.8 Give ha Fs () is he oe-side Laplace rasform of f, () he Laplace rasform of # f() τ dτ is (A) sf() s f() (B) () s Fs s (C) # F() τ dτ (D) [ ( ) ( )] s Fs f MCQ 6.9 A sysem wih rasfer fucio Hz () has impulse respose h (.) defied as h(), h() ad hk () oherwise. Cosider he followig saemes. S : Hz () is a low-pass filer. S : Hz () is a FIR filer. Which of he followig is correc? (A) Oly S is rue (B) Boh S ad S are false (C) Boh S ad S are rue, ad S is a reaso for S (D) Boh S ad S are rue, bu S is o a reaso for S Page 68 MCQ 6. Cosider a sysem whose ipu x ad oupu y are relaed by he equaio y () x ( τ) g( τ) dτ where h () is show i he graph. # Published by: NODIA ad COMPANY ISBN:

7 Which of he followig four properies are possessed by he sysem? BIBO : Bouded ipu gives a bouded oupu. Causal : The sysem is causal, LP : The sysem is low pass. LTI : The sysem is liear ad ime-ivaria. (A) Causal, LP (B) BIBO, LTI (C) BIBO, Causal, LTI (D) LP, LTI MCQ 6. The 4-poi Discree Fourier Trasform (DFT) of a discree ime sequece {,,,} is (A) [, + j,, j ] (B) [, + j, 6, j ] (C) [6, j,, + j ] (D) [6, + j,, j ] MCQ 6. s + A LTI sysem havig rasfer fucio ad ipu s + s+ x () si( + ) is i seady sae. The oupu is sampled a a rae ω s rad/s o obai he fial oupu { xk ( )}. Which of he followig is rue? (A) y (.) is zero for all samplig frequecies ω s (B) y (.) is ozero for all samplig frequecies ω s (C) y (.) is ozero for ω >, bu zero for < s (D) y (.) is zero for ω >, bu ozero for < s ω s ω 8 ONE MARK MCQ 6. The ipu ad oupu of a coiuous ime sysem are respecively deoed by x () ad y. () Which of he followig descripios correspods o a causal sysem? Published by: NODIA ad COMPANY ISBN: Page 69

8 (A) y ( ) x ( ) + x ( + 4) (B) y ( ) ( 4) x ( + ) (C) y ( ) ( + 4) x ( ) (D) y ( ) ( + 5) x ( + 5) MCQ 6.4 The impulse respose h () of a liear ime ivaria coiuous ime sysem is described by h () exp( αu ) () + exp( βu ) ( ) where u( ) deoes he ui sep fucio, ad α ad β are real cosas. This sysem is sable if (A) α is posiive ad β is posiive (B) α is egaive ad β is egaive (C) α is egaive ad β is egaive (D) α is egaive ad β is posiive 8 TWO MARKS MCQ 6.5 A liear, ime - ivaria, causal coiuous ime sysem has a raioal rasfer fucio wih simple poles a s ad s 4 ad oe simple zero a s. A ui sep u () is applied a he ipu of he sysem. A seady sae, he oupu has cosa value of. The impulse respose of his sysem is (A) [ exp( ) + exp( 4)] u( ) (B) [ 4 exp( ) exp( 4) exp( )] u( ) (C) [ 4 exp( ) + exp( 4)] u( ) (D) [ 5. exp( ) + 5. exp( 4)] u( ) Page 7 MCQ 6.6 The sigal x () is described by x () for # # + ) oherwise Two of he agular frequecies a which is Fourier rasform becomes zero are (A) π,π (B).5 π,.5π (C), π (D) π,.5π Published by: NODIA ad COMPANY ISBN:

9 MCQ 6.7 A discree ime liear shif - ivaria sysem has a impulse respose h [ ] wih h[ ], h[ ], h[ ], ad zero oherwise The sysem is give a ipu sequece x [ ] wih x[ ] x[ ], ad zero oherwise. The umber of ozero samples i he oupu sequece y, [ ] ad he value of y[ ] are respecively (A) 5, (B) 6, (C) 6, (D) 5, MCQ 6.8 Le x () be he ipu ad y () be he oupu of a coiuous ime sysem. Mach he sysem properies P, P ad P wih sysem relaios R, R, R, R4 Properies Relaios P : Liear bu NOT ime - ivaria R : y () x () P : Time - ivaria bu NOT liear R : y () x () P : Liear ad ime - ivaria R : y () x () (A) (P, R), (P, R), (P, R4) (B) (P, R), (P, R), (P, R4) (C) (P, R), (P, R), (P, R) (D) (P, R), (P, R), (P, R) R4 : y () x ( 5) MCQ 6.9 { x ( )} is a real - valued periodic sequece wih a period N. x ( ) ad Xk () form N-poi Discree Fourier Trasform (DFT) pairs. The N DFT Yk () of he sequece y ( ) xrx ( ) ( + r) N / is r (A) Xk () N (B) XrXk () ( + r) N N (C) XrXk () ( + r) N / (D) r / r Published by: NODIA ad COMPANY ISBN: Page 7

10 Saeme for Liked Aswer Quesio 6. ad 6.: I he followig ework, he swich is closed a ad he samplig sars from. The samplig frequecy is Hz. MCQ 6. The samples x ( ), (,,,...) are give by (A) 5( e 5. ) (B) 5e. 5 (C) 5( e 5 ) (D) 5e 5 MCQ 6. The expressio ad he regio of covergece of he z rasform of he sampled sigal are (A) 5z, z < e 5 (B) 5z 5 5., z < e. 5 z e z e (C) 5z 5., z > e 5. (D) 5z z e z e 5, z e > 5 Saeme for Liked Aswer Quesio 6. & 6.4: The impulse respose h () of liear ime - ivaria coiuous ime sysem is give by h () exp( ) u (), where u () deoes he ui sep fucio. MCQ 6. The frequecy respose H( ω ) of his sysem i erms of agular frequecy ω, is give by H( ω) (A) (B) si ω + jω ω (C) + jω (D) jω + jω Page 7 Published by: NODIA ad COMPANY ISBN:

11 MCQ 6. The oupu of his sysem, o he siusoidal ipu x ( ) cos for all ime, is 5. (A) (B) cos(.5 π) (C) cos(.5 π) (D) cos(.5 π) 7 ONE MARK MCQ 6.4 If he Laplace rasform of a sigal Ys (), he is fial ss ( ) value is (A) (B) (C) (D) Ubouded 7 TWO MARKS MCQ 6.5 The -db badwidh of he low-pass sigal e u(), where u () is he ui sep fucio, is give by (A) π Hz (B) Hz π (C) (D) Hz MCQ 6.6 A 5-poi sequece x [ ] is give as x[ ], x[ ], x[ ], x [] 5 ad x[ ]. Le Xe ( iω ) deoed he discree-ime Fourier π jω rasform of x. [ ] The value of Xe ( ) dω is # π (A) 5 (B) π (C) 6π (D) 5+ jπ MCQ The z rasform Xz () of a sequece x [ ] is give by Xz []. z I is give ha he regio of covergece of Xz () icludes he ui circle. The value of x[ ] is (A) 5. (B) (C).5 (D) 5 Published by: NODIA ad COMPANY ISBN: Page 7

12 MCQ 6.8 A Hilber rasformer is a (A) o-liear sysem (C) ime-varyig sysem (B) o-causal sysem (D) low-pass sysem MCQ 6.9 The frequecy respose of a liear, ime-ivaria sysem is give by 5 Hf () + jπf. The sep respose of he sysem is 5 (A) 5( e ) u( ) (B) 56 e 5@ u( ) (C) 5 ( e ) u( ) (D) ^ e u() 5 5h 6 ONE MARK MCQ 6.4 Le x ()* Xj ( ω ) be Fourier Trasform pair. The Fourier Trasform of he sigal x( 5 ) i erms of Xjω ( ) is give as jω (A) e X j ω jω 5 5 b 5 l (B) e X j ω 5 5 b 5 l (C) jω e X j ω 5 b 5 l (D) jω e X j ω 5 b 5 l MCQ 6.4 The Dirac dela fucio δ () is defied as (A) δ () ) oherwise (B) δ () ) oherwise (C) δ () ) ad δ () d oherwise # (D) δ () ) ad δ () d oherwise # Page 74 Published by: NODIA ad COMPANY ISBN:

13 MCQ 6.4 If he regio of covergece of x[ ] + x[ ] is < z < he he regio of covergece of x[ ] x[ ] icludes (A) < z < (B) < z < (C) < z < (D) < z < MCQ 6.4 I he sysem show below, x () ( si u ) () I seady-sae, he respose y () will be (A) si π ` 4 j (B) si π ` + 4 j (C) e si (D) si cos 6 TWO MARKS MCQ 6.44 Cosider he fucio f () havig Laplace rasform Fs () ω Re[] s > s + ω The fial value of f () would be (A) (B) (C) # f( ) # (D) MCQ 6.45 A sysem wih ipu x [ ] ad oupu y [ ] is give as y [ ] ( si 6 5 πx ) [ ]. The sysem is (A) liear, sable ad iverible (B) o-liear, sable ad o-iverible (C) liear, sable ad o-iverible (D) liear, usable ad iverible Published by: NODIA ad COMPANY ISBN: Page 75

14 MCQ 6.46 The ui sep respose of a sysem sarig from res is give by c () e for $. The rasfer fucio of he sysem is (A) (B) + s + s (C) (D) + s s + s MCQ 6.47 The ui impulse respose of a sysem is f () e, $. For his sysem he seady-sae value of he oupu for ui sep ipu is equal o (A) (B) (C) (D) 5 ONE MARK MCQ 6.48 Choose he fucio f (); cao be defied. < < for which a Fourier series (A) si( 5 ) (B) 4 cos( + ) + si( 7) (C) exp( ) si( 5) (D) MCQ 6.49 The fucio x () is show i he figure. Eve ad odd pars of a ui sep fucio u () are respecively, (A), x ( ) (B), x ( ) (C), x ( ) (D), x ( ) Page 76 Published by: NODIA ad COMPANY ISBN:

15 MCQ 6.5 The regio of covergece of z rasform of he sequece 5 u ( ) 6 b u( ) 6 l b5 l mus be (A) z < 5 (B) z 6 > 5 6 (C) 5 6 < z < 6 (D) 6 < z < 5 5 MCQ 6.5 Which of he followig ca be impulse respose of a causal sysem? MCQ 6.5 Ye j Le x ( ) ( ) u ( ), y ( ) x( ) ad ( ) be he Fourier rasform of y ( ) he ( ) (A) 4 (B) (C) 4 (D) 4 Ye jω MCQ 6.5 The power i he sigal s () 8cos( π π ) + 4si( 5π) is (A) 4 (B) 4 (C) 4 (D) 8 Published by: NODIA ad COMPANY ISBN: Page 77

16 5 TWO MARKS MCQ 6.54 The oupu y () of a liear ime ivaria sysem is relaed o is ipu x () by he followig equaios y () 5. x ( + T) + x ( ) + 5. x ( + T) d d d The filer rasfer fucio H( ω ) of such a sysem is give by (A) ( + cos ωte ) ω d (B) ( +.5 cos ωte ) ω d (C) ( cos ωte ) ω d (D) (.5 cos ωte ) ω d MCQ 6.55 Mach he followig ad choose he correc combiaio. Group E. Coiuous ad aperiodic sigal F. Coiuous ad periodic sigal G. Discree ad aperiodic sigal H. Discree ad periodic sigal Group. Fourier represeaio is coiuous ad aperiodic. Fourier represeaio is discree ad aperiodic. Fourier represeaio is coiuous ad periodic 4. Fourier represeaio is discree ad periodic (A) E,F,G4,H (B) E,F,G,H4 (C) E,F,G,H4 (D) E,F,G4,H Page 78 MCQ 6.56 A sigal x ( ) si( ω + φ) is he ipu o a liear ime- ivaria sysem havig a frequecy respose He ( jω ). If he oupu of he sysem Ax( ) he he mos geeral form of +H( e jω ) will be (A) ω + β for ay arbirary real (B) ω + πk for ay arbirary ieger k (C) ω + πk for ay arbirary ieger k (D) ωφ Published by: NODIA ad COMPANY ISBN:

17 Saeme of liked aswer quesio 6.59 ad 6.6 : A sequece x ( ) has o-zero values as show i he figure. MCQ 6.57 The sequece y ( ) x( ), For eve * will be, For odd Published by: NODIA ad COMPANY ISBN: Page 79

18 MCQ 6.58 The Fourier rasform of y( ) will be (A) e j ω [ cos 4ω+ cos ω+ ] (B) cos ω+ cos ω+ (C) e j ω [ cos ω+ cos ω+ ] (D) e j ω [ cos ω + cos + ] MCQ 6.59 For a sigal x () he Fourier rasform is Xf. () The he iverse Fourier rasform of X( f+ ) is give by (A) x jπ e `j (B) x - e `j (C) x( ) e j4π (D) x ( + ) j4π 4 ONE MARK MCQ 6.6 The impulse respose h [ ] of a liear ime-ivaria sysem is give by h [ ] u [ + ] + u [ ) [ 7] where u [ ] is he ui sep sequece. The above sysem is (A) sable bu o causal (C) causal bu usable (B) sable ad causal (D) usable ad o causal MCQ 6.6 z The z -rasform of a sysem is Hz () z.. If he ROC is z <., he he impulse respose of he sysem is (A) (.) u [] (B) (.) u[ ] (C) (.) u [] (D) (.) u[ ] MCQ 6.6 Page 8 The Fourier rasform of a cojugae symmeric fucio is always (A) imagiary (C) real (B) cojugae ai-symmeric (D) cojugae symmeric Published by: NODIA ad COMPANY ISBN:

19 4 TWO MARKS MCQ 6.6 Cosider he sequece x [ ] [ 4 j5+ j5]. The cojugae aisymmeric par of he sequece is - (A) [ 4 j.5, j, 4 j.5] (B) [ j.5,, j.5] (C) [ j.5, j, ] (D) [ 4,, 4] MCQ 6.64 A causal LTI sysem is described by he differece equaio y [ ] αy [ ] x [ ] + βx [ ] The sysem is sable oly if (A) α, β < (B) α >, β > (C) α <, ay value of β (D) β <, ay value of α MCQ 6.65 The impulse respose h [ ] of a liear ime ivaria sysem is give as h [ ] *, 4, oherwise If he ipu o he above sysem is he sequece e oupu is jπ/ 4 jπ/ 4 (A) 4 e (B) 4 e (C) 4e jπ/ 4 (D) 4e j π / 4 jπ/ 4, he he MCQ 6.66 Le x () ad y () wih Fourier rasforms Ff () ad Yf () respecively be relaed as show i Fig. The Yf () is Published by: NODIA ad COMPANY ISBN: Page 8

20 j π f (A) Xf ( /) e (C) Xf (/ ) e jπf (B) Xf (/ ) e jπf (D) Xf ( /) e j π f ONE MARK MCQ 6.67 The Laplace rasform of i () is give by Is () s( + s) A ", The value of i () eds o (A) (B) (C) (D) MCQ 6.68 The Fourier series expasio of a real periodic sigal wih fudameal frequecy f is give by gp () ce j π / f. I is give ha c + j5 -. The c is (A) 5+ j (B) j5 (C) 5+ j (D) j5 MCQ 6.69 Le x () be he ipu o a liear, ime-ivaria sysem. The required oupu is 4π( ). The rasfer fucio of he sysem should be (A) 4e j4π f (B) e j8πf (C) 4e j 4πf (D) e j8πf Page 8 MCQ A sequece x ( ) wih he z rasform Xz () z+ z z+ z is applied as a ipu o a liear, ime-ivaria sysem wih he impulse respose h ( ) δ( ) where, δ ( ) ), oherwise The oupu a 4 is (A) 6 (B) zero (C) (D) 4 Published by: NODIA ad COMPANY ISBN:

21 TWO MARKS MCQ 6.7 Le P be lieariy, Q be ime-ivariace, R be causaliy ad S be sabiliy. A discree ime sysem has he ipu-oupu relaioship, x ( ) $ y ( ) *, x ( + ) # where x ( ) is he ipu ad y ( ) is he oupu. The above sysem has he properies (A) P, S bu o Q, R (B) P, Q, S bu o R (C) P, Q, R, S (D) Q, R, S bu o P Commo daa for Q 6.7 & 6.74 : The sysem uder cosideraio is a RC low-pass filer (RC-LPF) wih R kω ad C. μf. MCQ 6.7 Le Hf () deoe he frequecy respose of he RC-LPF. Le f be he Hf () highes frequecy such ha # f # f $ 95.. The f H() (i Hz) is (A) 4.8 (B) 6.9 (C) 5. (D) 4.4 MCQ 6.7 Le g () f be he group delay fucio of he give RC-LPF ad f Hz. The g () f i ms, is (A).77 (B) 7.7 (C) 7.7 (D) 4.55 ONE MARK MCQ 6.74 Covoluio of x ( + 5) wih impulse fucio δ( 7) is equal o (A) x ( ) (B) x ( + ) (C) x ( ) (D) x ( + ) Published by: NODIA ad COMPANY ISBN: Page 8

22 MCQ 6.75 Which of he followig cao be he Fourier series expasio of a periodic sigal? (A) x ( ) cos + cos (B) x ( ) cos π+ 7cos (C) x () cos + 5. (D) x () cos. 5π+ si. 5π MCQ 6.76 The Fourier rasform F{ e u( )} is equal o F ' + jπ is f f (A) euf () (B) e u() f f (C) eu( f) (D) e f u( f) + jπf. Therefore, MCQ 6.77 A liear phase chael wih phase delay T p ad group delay T g mus have (A) Tp Tg cosa (B) Tp \ f ad Tg \ f (C) T p cosa ad Tg \ f (f deoe frequecy) (D) Tp \ f ad T p cosa TWO MARKS MCQ 6.78 The Laplace rasform of coiuous - ime sigal x () is (). If he Fourier rasform of his sigal exiss, he x () is (A) e u() e u() (B) e u( ) + e u( ) (C) e u( ) e u( ) (D) e u( ) e u( ) 5s Xs s s MCQ 6.79 If he impulse respose of discree - ime sysem is h [ ] 5 u[ ], he he sysem fucio Hz () is equal o (A) z ad he sysem is sable z 5 Page 84 Published by: NODIA ad COMPANY ISBN:

23 (B) z z 5 (C) z z 5 (D) z z 5 ad he sysem is sable ad he sysem is usable ad he sysem is usable ONE MARK MCQ 6.8 The rasfer fucio of a sysem is give by Hs () impulse respose of he sysem is (A) ( * e ) u( ) (B) ( * e ) u( ) (C) ( e ) u( ) (D) ( e ) u( ) s ( s ). The MCQ 6.8 The regio of covergece of he z rasform of a ui sep fucio is (A) z > (B) z < (C) (Real par of z ) > (D) (Real par of z ) < MCQ 6.8 Le δ () deoe he dela fucio. The value of he iegral ()cos # δ b d l is (A) (B) (C) (D) π MCQ 6.8 If a sigal f () has eergy E, he eergy of he sigal f( ) is equal o (A) (B) E/ (C) E (D) 4E Published by: NODIA ad COMPANY ISBN: Page 85

24 TWO MARKS MCQ 6.84 The impulse respose fucios of four liear sysems S, S, S, S4 are give respecively by h (), h () u(), u () h () ad + h () e u() 4 where u () is he ui sep fucio. Which of hese sysems is ime ivaria, causal, ad sable? (A) S (B) S (C) S (D) S4 ONE MARK MCQ 6.85 Give ha Lf [()] s +, [ ( )] Lg s + s + ( s + )( s + ) ad h () # f( τ) g ( τ) dτ. Lh [ ( )] is (A) s + s + (C) s + + s + ( s + )( s + ) s + (B) s + (D) Noe of he above MCQ 6.86 The Fourier Trasform of he sigal x () e is of he followig form, where A ad B are cosas : (A) Ae Bf (B) Ae Bf (C) A+ B f (D) Ae Bf Page 86 Published by: NODIA ad COMPANY ISBN:

25 MCQ 6.87 A sysem wih a ipu x () ad oupu y () is described by he relaios : y() x(). This sysem is (A) liear ad ime - ivaria (B) liear ad ime varyig (C) o - liear ad ime - ivaria (D) o - liear ad ime - varyig MCQ 6.88 A liear ime ivaria sysem has a impulse respose e, >. If he iiial codiios are zero ad he ipu is e, he oupu for > is (A) e (C) e e (B) e 5 + e (D) Noe of hese TWO MARKS MCQ 6.89 Oe period (, T) each of wo periodic waveforms W ad W are show i he figure. The magiudes of he h Fourier series coefficies of W ad W, for $, odd, are respecively proporioal o (A) ad (B) ad (C) ad (D) 4 ad MCQ 6.9 Le u () be he sep fucio. Which of he waveforms i he figure correspods o he covoluio of u () u ( ) wih u () u ( )? Published by: NODIA ad COMPANY ISBN: Page 87

26 MCQ 6.9 A sysem has a phase respose give by φω ( ), where ω is he agular frequecy. The phase delay ad group delay a ω ω are respecively give by φω ( ) dφω ( ) (A), ω dω (C) ωo ( ), dφω ( ) φω d( ω) o ω ω ω ω o (B) φω ( ), d φω ( ) dω o o (D) ωφ( ω), φ( λ) o o # ω ω ω o 999 ONE MARK MCQ 6.9 The z -rasform Fz () of he fucio ft ( ) a T is (A) z z a T (B) z z a T + (C) z z a T (D) z z a MCQ T If [ f ()] Fs (), he [( f T)] is equal o st st (A) e F() s (B) e F() s Fs () Fs () (C) st (D) e e st Page 88 Published by: NODIA ad COMPANY ISBN:

27 MCQ 6.94 A sigal x () has a Fourier rasform X( ω ). If x () is a real ad odd fucio of, he X( ω ) is (A) a real ad eve fucio of ω (B) a imagiary ad odd fucio of ω (C) a imagiary ad eve fucio of ω (D) a real ad odd fucio of ω 999 TWO MARKS MCQ 6.95 The Fourier series represeaio of a impulse rai deoed by s () / d ( T) is give by (A) jπ / exp (B) jπ / exp T T T T (C) T jπ / exp (D) jπ / exp T T T MCQ 6.96 The z -rasform of a sigal is give by Cz () 4 z ( z ) 4 ( z ) Is fial value is (A) /4 (C). (B) zero (D) ifiiy 998 ONE MARK MCQ 6.97 If Fs () ω, he he value of Limf() s + ω " (A) cao be deermied (C) is uiy (B) is zero (D) is ifiie Published by: NODIA ad COMPANY ISBN: Page 89

28 MCQ 6.98 The rigoomeric Fourier series of a eve ime fucio ca have oly (A) cosie erms (B) sie erms (C) cosie ad sie erms (D) d.c ad cosie erms MCQ 6.99 A periodic sigal x () of period T is give by, < T x () *, T < < T The dc compoe of x () is (A) T (B) T T T (C) T (D) T T T MCQ 6. The ui impulse respose of a liear ime ivaria sysem is he ui sep fucio u. () For >, he respose of he sysem o a a exciaio e u(), a > will be (A) ae a (B) ( /)( a e a ) a (C) a( e ) (D) e a MCQ 6. The z-rasform of he ime fucio / δ( k) is (A) z z k (B) z z (C) z ( z ) (D) ( z ) z MCQ 6. A disored siusoid has he ampliudes A, A, A,... of he fudameal, secod harmoic, hird harmoic,... respecively. The oal harmoic disorio is (A) A+ A+... (B) A + A +... A A Page 9 Published by: NODIA ad COMPANY ISBN:

29 (C) A + A +... A + A + A +... (D) A c + A +... A m MCQ 6. The Fourier rasform of a fucio x () is Xf. () The Fourier rasform dx() of will be df (A) dx() f df (C) jfx() f (B) jπfx() f (D) Xf () jf 997 ONE MARK MCQ 6.4 The fucio f () has he Fourier Trasform g( ω ). The Fourier Trasform jω ffg () () e # ge () do is (A) f( ) π ω (B) f( ω π ) (C) πf( ω) (D) Noe of he above MCQ 6.5 α The Laplace Trasform of e cos( α) is equal o ( s α) ( s + α) (A) ( s ) (B) α + α ( s α ) + α (C) ( s α ) (D) Noe of he above 996 ONE MARK MCQ 6.6 The rigoomeric Fourier series of a eve fucio of ime does o have he (A) dc erm (B) cosie erms (C) sie erms (D) odd harmoic erms Published by: NODIA ad COMPANY ISBN: Page 9

30 MCQ 6.7 The Fourier rasform of a real valued ime sigal has (A) odd symmery (B) eve symmery (C) cojugae symmery (D) o symmery *********** Page 9 Published by: NODIA ad COMPANY ISBN:

31 SOLUTIONS SOL 6. We have dy dy + y x () d d Applyig Laplace rasform we ge sys () sys () + Ys () Xs () Ys () or Hs () Xs () s s+ / A s (/) 5s / + s + ξω s + ω Here ω / ad ξω / 5 givig ξ Roos are s /, / which lie o Righ side of s plae hus usable. Hece (A) is correc opio. SOL 6. For a eve fucio Fourier series coais dc erm ad cosie erm (eve ad odd harmoics). Hece (C) is correc opio. SOL 6. Fucio h ( ) au ( ) sable if a < ad Usable if a H We We have h ( ) u ( ) ; Here a herefore h ( ) is usable ad sice h ( ) for < Therefore h ( ) will be causal. So h ( ) is causal ad o sable. Hece (B) is correc opio. SOL 6.4 Impulse respose d ( sep respose) d d ( e ) d α α + αe αe Hece (A) is correc opio. Published by: NODIA ad COMPANY ISBN: α Page 9

32 SOL 6.5 We have x () exp( ) μ( ) + s( 6) ad h () u () Takig Laplace Trasform we ge Xs () b e s + 6s + l ad Hs () s Now Ys () HsXs () () 6s e e s s+ + ss ( + ) + : D s or Ys () e s ( s + ) + s Thus y ().5[ exp( )] u( ) + u( 6) Hece (D) is correc opio. 6s 6s SOL 6.6 y ( ) x ( ) or Yz () z X() z Yz () or Hz () z Xz () Now H() z H () z z. 4z c H() z. 6z m z z (. 6z ) H () z (. 4z ) Hece (B) is correc opio. SOL 6.7 * * * For 8 poi DFT, x [ ] x[ 7]; x [ ] x[ 6]; x [ ] x[ 5] ad i is cojugae symmeric abou x[ 4 ], x[ 6 ] ; x[ 7] + j Hece (B) is correc opio. SOL 6.8 Page 94 For a fucio x () rigoomeric fourier series is x () A + [ A cos ω+ B si ω] / o Where, A o () T # xd T "fudameal period T ad A x ()cosωd T # T Published by: NODIA ad COMPANY ISBN:

33 B x ()siωd T # T For a eve fucio x (), B Sice give fucio is eve fucio so coefficie B, oly cosie ad cosa erms are prese i is fourier series represeaio Cosa erm A T/ 4 xd () T # T/ 4 Cosa erm is egaive. Hece (C) is correc opio. T/ 4 T/ 4 Ad Ad T :# + # D T/ 4 T/ 4 T : TA A T D A SOL 6.9 Iverse Z rasform We kow ha α Z a αδ[! a] Give ha Xz () 5z + 4z + Iverse z-rasform x [ ] 5δ[ + ] + 4δ[ ] + δ[ ] Hece (A) is correc opio. SOL 6. We have h [ ] δ[ ] or H[ Z] Z ad h [ ] δ[ ] or H( Z) Z Respose of cascaded sysem Hz () H() z : H() z z : z z or, h [ ] δ[ ] Hece (C) is correc opio. SOL 6. For a N-poi FET algorihm buerfly operaes o oe pair of samples ad ivolves wo complex addiio ad oe complex muliplicaio. Hece (D) is correc opio. SOL 6. We have f () ad lim f () " L s + ; s + 4s + ( k) s E Published by: NODIA ad COMPANY ISBN: Page 95

34 By fial value heorem or or lim f () " s.( s+ ) lim s s 4 s " + + ( k ) s s( s+ ) lim s " ss [ + 4 s + ( k )] k or k 4 Hece (D) is correc opio. lim sf() s s " SOL 6. Sysem is described as d y() d() dx() y () + 4x () d d d Takig laplace rasform o boh side of give equaio sys () + 4sYs () + Ys () sx() s + 4X() s ( s + 4s+ ) Y( s) ( s+ ) X( s) s Trasfer fucio of he sysem Ys () ( s + ) ( s + ) Hs () Xs () s + 4s + ( s + )( s + ) Ipu x () e u() or, Xs () ( s + ) Oupu Ys () Hs (): Xs () ( s + ) Ys () : ( s + )( s + ) ( s + ) By Parial fracio Ys () s + s + Takig iverse laplace rasform y () ( e e ) u( ) Hece (B) is correc opio. SOL 6.4 Page 96 We have Hz () 4 z z + 4 Published by: NODIA ad COMPANY ISBN: z 8

35 By parial fracio Hz () ca be wrie as Hz () + z ^ h ^ 4 z h For ROC : z > / h [ ] u [ ] u [ ], b > au [ ], z > a l + b 4 l z Thus sysem is causal. Sice ROC of Hz () icludes ui circle, so i is sable also. Hece S is True For ROC : z < 4 h [ ] u[ ] u ( ), z, z b > < l + b 4 l 4 Sysem is o causal. ROC of Hz () does o iclude uiy circle, so i is o sable ad S is True Hece (C) is correc opio. SOL 6.5 The Fourier series of a real periodic fucio has oly cosie erms if i is eve ad sie erms if i is odd. Hece (A) is correc aswer. SOL 6.6 Give fucio is f () si + cos cos + cos + cos The fucio has a DC erm ad a cosie fucio. The frequecy of cosie erms is ω πf " f Hz π The give fucio has frequecy compoe a ad π Hz. Hece (B) is correc aswer. SOL 6.7 x [ ] u ( ) b u( ) l b l Takig z rasform we have Xz () z / b z l / b l Published by: NODIA ad COMPANY ISBN: Page 97

36 Firs erm gives Secod erm gives z z / / b z l < " < > " > z z b z Thus is ROC is he commo ROC of boh erms. ha is < z < Hece (A) is correc aswer. l SOL 6.8 By propery of uilaeral laplace rasform L Fs () f() d # τ τ + f() τ dτ s s # Here fucio is defied for < τ <, Thus L Fs () # f() τ s Hece (B) is correc aswer. SOL 6.9 We have h(), h() oherwise hk (). The diagram of respose is as follows : I has he fiie magiude values. So i is a fiie impulse respose filer. Thus S is rue bu i is o a low pass filer. So S is false. Hece (A) is correc aswer. SOL 6. Page 98 Here h ()! for <. Thus sysem is o causal. Agai ay bouded ipu x () gives bouded oupu y. () Thus i is BIBO sable. Here we ca coclude ha opio (B) is correc. Hece (B) is correc aswer. Published by: NODIA ad COMPANY ISBN:

37 SOL 6. We have x [ ] {,,, ) ad N 4 N jπk/ N Xk [] / xe [ ] k,... N jπk/ 4 For N 4, Xk [] / xe [ ] k,,... / Now X[ ] x [ ] x[ ] + x[ ] + x[ ] + x[ ] / x[ ] xe [ ] jπ/ x[] + x[] e + x[] e + x[] e + + j + j / X[ ] xe [ ] j π jπ / jπ jπ / x[] + x[] e + x[] e + x[] e + + / X[ ] xe [ ] jπ/ j π j π j π x[] + x[] e + x[] e + x[] e + j j Thus [ 6, + j,, j] Hece (D) is correc aswer. jπ / jπ j9π / SOL 6. Hece (A) is correc aswer. SOL 6. The oupu of causal sysem depeds oly o prese ad pas saes oly. I opio (A) y() depeds o x( ) ad x() 4. I opio (B) y() depeds o x(). I opio (C) y() depeds o x( ). I opio (D) y() depeds o x() 5. Thus oly i opio (C) he value of y () a depeds o x( ) pas value. I all oher opio prese value depeds o fuure value. Hece (C) is correc aswer. Published by: NODIA ad COMPANY ISBN: Page 99

38 SOL 6.4 α β We have h () e u() + e u( ) This sysem is sable oly whe bouded ipu has bouded oupu For sabiliy α < for > ha implies α < ad β > for > ha implies β >. Thus, α is egaive ad β is posiive. Hece (D) is correc aswer. SOL 6.5 Gs () Ks ( + ) ( s + )( s + 4), ad Rs () s Ks ( + ) Cs () GsRs () () ss ( + )( s+ 4) K + K K 8s 4( s + ) 8( s + 4) Thus c () K e e u() : + 4 D A seady-sae, c( ) Thus K or K 8 8 The, Gs () 8( s + ) ( s + )( s + 4) 4 ( s + 4) ( s + ) h () L G() s ( 4e 4 + e ) u( ) Hece (C) is correc aswer. SOL 6.6 Page 4 for # # + We have x () ) oherwise Fourier rasform is e j ω jω x() d e d # # j [ e ] ω jω jω jω ( e e ) ( j si ω) jω jω si ω ω This is zero a ω π ad ω π Hece (A) is correc aswer. Published by: NODIA ad COMPANY ISBN:

39 SOL 6.7 Give h ( ) [,, ] x ( ) [,, ] y ( ) x ( )* h ( ) The legh of y [ ] is L + L + 5 / k y ( ) x ( )* h ( ) xkh ( ) ( k) / k y() xkh () ( k) x( ) h( ) + x( ) h( ) + x( ) h( ) h() + + h() + There are 5 o zero sample i oupu sequece ad he value of y[ ] is. Hece (D) is correc aswer. SOL 6.8 Mode fucio are o liear. Thus y () x () is o liear bu his fucios is ime ivaria. Opio (A) ad (B) may be correc. The y () x () is o liear, hus opio (B) is wrog ad (a) is correc. We ca see ha R: y( ) x( ) Liear ad ime varia. R : y( ) x( ) No liear ad ime varia. R : y( ) x ( ) No liear ad ime ivaria R4 : y( ) x( 5) Liear ad ime ivaria Hece (B) is correc aswer. SOL 6.9 N - / Give : y ( ) xrx () ( + r) N r I is Auo correlaio. Hece y ( ) rxx ( ) X( k) Hece (A) is correc aswer. DFT SOL 6. Curre hrough resisor (i.e. capacior) is + I I( ) e RC / Here, I( + ) V 5 5μA R k Published by: NODIA ad COMPANY ISBN: Page 4

40 RC k # μ sec I 5e μa VR # R 5e V Here he volages across he resisor is ipu o sampler a frequecy of Hz. Thus x ( ) 5e 5e 5. # For > Hece (B) is correc aswer. SOL 6. Sice x ( ) 5 e 5. u( ) is a causal sigal Is z rasform is Xz () 5 : 5. e z D 5z. z 5 e Is ROC is e z > " z > e Hece (C) is correc aswer SOL 6. h () e u() jω d # ω # # Hjω ( ) he () Hece (C) is correc aswer. e ( ) e j + d e j ω d ( + jω) SOL 6. Page 4 Hjω ( ) ( + jω) The phase respose a ω rad/sec is + Hj ( ω ) a ω a π.5π 4 Magiude respoe a ω rad/sec is Hjω ( ) + w Ipu is x () cos( ) Oupu i # cos(. 5π) cos[. 5π] Hece (D) is correc aswer. Published by: NODIA ad COMPANY ISBN:

41 SOL 6.4 Ys () ss ( ) Fial value heorem is applicable oly whe all poles of sysem lies i lef half of S -plae. Here s is righ s plae pole. Thus i is ubouded. Hece (D) is correc aswer. SOL 6.5 x () e u() Takig Fourier rasform Xjω ( ) + jω Xjω ( ) + ω Magiude a db frequecy is Thus + ω or ω rad or f Hz π Hece (A) is correc aswer. SOL 6.6 For discree ime Fourier rasform (DTFT) whe N " π x [ ] Xe ( ) e d j ω π j ω ω π Puig we ge x[ ] Xe ( ) e d π j ω π j ω ω # Xe ( ) d π π π j ω ω # π π jω or Xe ( ) dω x[ ] # π π# 5 π π Hece (B) is correc aswer. # SOL 6.7 Xz () 5. z Sice ROC icludes ui circle, i is lef haded sysem Published by: NODIA ad COMPANY ISBN: Page 4

42 x ( ) (.5)() u( ) x() If we apply iiial value heorem x() lim Xz () lim 5. z " z " z.5 Tha is wrog because here iiial value heorem is o applicable because sigal x ( ) is defied for <. Hece (B) is correc aswer. SOL 6.8 A Hilber rasformer is a o-liear sysem. Hece (A) is correc aswer. SOL 6.9 Hf () 5 + jπf Hs () s 5^s+ 5 h s + Sep respose Ys () a s ^s+ 5 h or Ys () 5 5 s s 5 s ^ + h s + 5 / or y () 5( e 5 ) u( ) Hece (B) is correc aswer. 5 SOL 6.4 x () Xj ( ω) Usig scalig we have F x( 5) X j ω 5 c 5 m Usig shifig propery we ge x 5 X j e 5 F ω ; b le 5 b 5 l Hece (A) is correc aswer. F jω 5 SOL 6.4 Page 44 Dirac dela fucio δ () is defied a ad i has ifiie value a. The area of dirac dela fucio is uiy. Hece (D) is correc opio. Published by: NODIA ad COMPANY ISBN:

43 SOL 6.4 The ROC of addiio or subracio of wo fucios x ( ) ad x ( ) is R+ R. We have bee give ROC of addiio of wo fucio ad has bee asked ROC of subracio of wo fucio. I will be same. Hece (D) is correc opio. SOL 6.4 As we have x () si, hus ω Now Hs () s + or Hjω ( ) jω + j+ or Hjω ( ) + 45c Thus y () si( π 4 ) Hece (A) is correc opio. SOL 6.44 Fs () ω + ω L F() s si ω o f () si s ω o Thus he fial value is # f( ) # Hece (C) is correc aswer. SOL 6.45 y ( ) si 5 b π x( ) 6 l Le x ( ) δ( ) Now y ( ) si (bouded) BIBO sable Hece (C) is correc aswer. SOL 6.46 c () e Takig laplace rasform Cs () Cs () # s Us () ss ( + ) s + Hece (B) is correc aswer. Published by: NODIA ad COMPANY ISBN: Page 45

44 SOL 6.47 L h () e H() s s + x () u () Xs () L Ys () HsXs () () # s + s s s+ y () u () e I seady sae i.e. ", y( ) Hece (C) is correc aswer. s SOL 6.48 Fourier series is defied for periodic fucio ad cosa. si( 5 ) is a periodic fucio. 4 cos( + ) + si( 7) is sum of wo periodic fucio ad also a periodic fucio. e si(5 ) is o a periodic fucio, so FS ca be defied for i. is cosa Hece (C) is correc opio. SOL 6.49 Ev{ g ( )} odd{ g ( )} g () + g( ) g () g( ) Here g () u () Thus u () + u( ) ue () Hece (A) is correc aswer. SOL 6.5 u () u( ) x () uo () Page 46 5 Here x ( ) ` u ( ) 6 j X () z 5 ^6 z h x ( ) 6 ` u( ) 5 j ROC : R Published by: NODIA ad COMPANY ISBN: " z > 5 6

45 X () z 6 ROC : R ^5 z h Thus ROC of x( ) + x( ) is R+ R which is 5 < z < Hece (C) is correc aswer. " z < 6 5 SOL 6.5 For causal sysem h () for #. Oly (D) saisfy his codiio. Hece (D) is correc aswer. SOL 6.5 x ( ) b u ( ) l y ( ) x ( ) b u ( ) l or y ( ) u ( ) ; b u ( ) l E b 4 l...() Ye ( jω ) ye ( ) j ω / j ω / b e 4 l or Ye ( j ) / `4 j + b4 l + b l b4 l b4 l or Ye ( j ) Aleraive : Takig z rasform of () we ge Yz () z Subsiuig z ω we have Ye ( jω ) j ω 4 e Ye ( j ) e j Hece (D) is correc aswer SOL 6.5 s () 8 cos π ` π + 4 si 5π j 8si π+ 4si 5π Here A 8 ad A 4. Thus power is Published by: NODIA ad COMPANY ISBN: Page 47

46 P A + A Hece (A) is correc aswer SOL 6.54 y () 5. x ( d+ T) + x ( d) + 5. x ( dt) Takig Fourier rasform we have Y( ω ).5 e j ω ( d+ T) X( ) e j ω d X( ).5 e j ω ( d ω + ω + T) X( ω) Y( ω) jωd jωt jωt or e [.5e e ] X( ω) e j d ω [.5( e j ω T + e j ω T jω ) + ] e d [ cos ωt+ ] Y( ω) jω or H( ω ) e d ( cos ωt+ ) X( ω) Hece (A) is correc aswer. SOL 6.55 For coiuous ad aperiodic sigal Fourier represeaio is coiuous ad aperiodic. For coiuous ad periodic sigal Fourier represeaio is discree ad aperiodic. For discree ad aperiodic sigal Fourier represeaio is coiuous ad periodic. For discree ad periodic sigal Fourier represeaio is discree ad periodic. Hece (C) is correc aswer. SOL 6.56 Page 48 y ( ) Ax( o ) Takig Fourier rasform Ye ( jω ) Ae j ωo o X( e j ω ) or He ( jω jω Ye ( ) ) Xe ( jω ) Ae jω Thus +H( e jω ) ω o o For LTI discree ime sysem phase ad frequecy of He ( jω ) are periodic wih period π. So i geeral form θω ( ) oωo+ πk Hece (B) is correc aswer. Published by: NODIA ad COMPANY ISBN: o o

47 SOL 6.57 From x ( ) [,,,,, ] y ( ) x^ h, eve, for odd, y( ) x( ) x( ), y( ), y() x( ) x( ), y() y() x( ) x( ), y() 4 y() 4 x( 4 ) x( ) 5, y() 5 6 y() 6 x( 6 ) x( ) Hece y ( ) δ( + ) + δ( ) + δ( ) + δ( 4) + δ( 6) Thus (A) is correc opio. SOL 6.58 Here y ( ) is scaled ad shifed versio of x ( ) ad agai y( ) is scaled versio of y ( ) givig z ( ) y( ) x( ) δ( + ) + δ( ) + δ( ) + δ( ) + δ( ) Takig Fourier rasform. Ze ( ) e + + e + e + e jω jω jω jω e e e e b e e jω jω jω jω jω jω e b + e jω jω jω + e + + e or Ze ( jω ) e jω [ cos ω+ cos ω+ ] Hece (C) is correc aswer. jω l l SOL 6.59 x () Xf () Usig scalig we have F xa ( ) a X f c a m F Published by: NODIA ad COMPANY ISBN: Page 49

48 Thus x F b f l X( f) Usig shifig propery we ge jπf e x() Xf ( + f ) Thus e j 4 π x b l F X ( f+ ) Hece (B) is correc aswer. jπ e x F b l X( ( f+ )) 4 jπ e x F b l X[( f+ )] SOL 6.6 A sysem is sable if / h ( ) <. The plo of give h ( ) is 6 Thus / h ( ) / h ( ) < Hece sysem is sable bu h ( )! for <. Thus i is o causal. Hece (A) is correc aswer. SOL 6.6 Page 4 Hz () z z <. z. We kow ha au[ ] * az z < a Thus h [ ] (.) u[ ] Hece (D) is correc aswer. Published by: NODIA ad COMPANY ISBN:

49 SOL 6.6 The Fourier rasform of a cojugae symmerical fucio is always real. Hece (C) is correc aswer. SOL 6.6 We have x ( ) [ 4 j5, + j, 4] x*( ) [ 4 + j5, j, 4] x*( ) [4, j, 4 + j5] x cas x ( ) x( ) ( ) Hece (A) is correc aswer. - * [ 4 j, j 4 j ] SOL 6.64 We have y ( ) αy ( ) x ( ) + βx ( ) Takig z rasform we ge Yz () Yzz () α Xz () + βxzz () Yz () βz or Xz () c m...(i) αz or Hz () β z( z) α ( z ) I has poles a! α / ad zero a ad β /. For a sable sysem poles mus lie iside he ui circle of z plae. Thus α < or α < Bu zero ca lie aywhere i plae. Thus, β ca be of ay value. Hece (C) is correc aswer. SOL 6.65 j / 4 π We have x ( ) e ad h ( ) 4 δ( + ) δ( + ) δ( ) + 4 δ( ) Now y ( ) x ( )* h ( ) Published by: NODIA ad COMPANY ISBN: Page 4

50 / x ( khk ) ( ) / x ( khk ) ( ) k k or y ( ) x ( + ) h( ) + x ( + ) h( ) + x ( ) h() + x ( ) h() 4 e e e + 4 e π ( ) π ( ) π j j j ( ) π + + j ( ) e + 6e + e 4 e e e e e e π 6 π 4 π π j j 4 4 π e [] e [ cos 4] or y ( ) 4e j r Hece (D) is correc aswer. π ( ) π ( ) π j j j ( ) π + + j ( ) SOL 6.66 From give graph he relaio i x () ad y () is y () x[ ( + )] Usig scalig we have F x () Xf () xa ( ) F a X f c a m F Thus x( ) X f c m Usig shifig propery we ge jπf x ( ) e X() f Thus x[ ( + )] e X f jπf F jπf( ) e X f bl bl x[ ( + )] Hece (B) is correc aswer. F j f e π X f c m SOL 6.67 From he Fial value heorem we have lim i () " lim si() s lim s lim s " s s ( + s ) s ( + s ) " " Page 4 Hece (C) is correc aswer. Published by: NODIA ad COMPANY ISBN:

51 SOL 6.68 Here C + j5 For real periodic sigal C k C k * Thus C Ck j5 Hece (D) is correc aswer. SOL 6.69 y () 4x ( ) Takig Fourier rasform we ge Ye ( jπf ) 4 jπf e X ( jπf e ) jπf Ye ( ) or 4e 4j π f jπf Xe ( ) Thus He ( jπf ) 4e 4j π f Hece (C) is correc aswer. Time Shifig propery SOL 6.7 We have h ( ) δ( ) or Hz () z Takig z rasform 4 4 Xz () z + z z+ z Now Yz () HzXz () () z ( z z + z+ z ) ( z z z z + + z ) Takig iverse z rasform we have y ( ) [ δ( + ) + δ( ) δ( ) + δ( ) δ( 7)] A 4, y() 4 Hece (B) is correc aswer. SOL 6.7 Sysem is o causal because oupu depeds o fuure value For # y( ) x( + ) x( ) y ( ) x ( + ) Time varyig y ( ) x ( + ) Depeds o Fuure i.e. y() x() Noe causal Published by: NODIA ad COMPANY ISBN: Page 4

52 For bouded ipu, sysem has bouded oupu. So i is sable. y ( ) x ( ) for $ for So sysem is liear. Hece (A) is correc aswer. xx ( + ) for # SOL 6.7 The frequecy respose of RC-LPF is Hf () + jπfrc Now H() Hf () H() or or + 4π f R C #. 8 4π f R C #. 8 or π frc #. 9 or f. # 9 π RC or f or f. + 4π f R C $ 95 #. 9 π RC #. 9 π k # μ or f # 5. Hz Thus f max 5. Hz Hece (C) is correc aswer. SOL 6.7 H( ω ) + jωrc θω ( ) a ωrc ( ) g dθω RC dω + ω RC π # # ms Page 44 Hece (A) is correc aswer. Published by: NODIA ad COMPANY ISBN:

53 SOL 6.74 If x ()* h () g () The x ( τ)* h ( τ) y ( ττ) Thus x ( + 5)* δ( 7) x ( + 5 7) x ( ) Hece (C) is correc aswer. SOL 6.75 I opio (B) he give fucio is o periodic ad does o saisfy Dirichle codiio. So i ca be expasio i Fourier series. x () cos π+ 7cos T π ω T π π T irraioal T π Hece (B) is correc aswer. SOL 6.76 From he dualiy propery of fourier rasform we have FT If x () Xf () The X () x( f) FT Therefore if e u() + jπf FT The + jπ Hece (C) is correc aswer. FT f eu( f) SOL 6.77 ad θω ( ) ω θω ( ) p ω g dθω ( ) dω Thus p g cosa Hece (A) is correc aswer. Published by: NODIA ad COMPANY ISBN: Page 45

54 SOL 6.78 Xs () 5 s s s 5 s ( s + )( s ) + s + s Here hree ROC may be possible. Re ( s ) < Re ( s ) > < Re ( s ) < Sice is Fourier rasform exis, oly < Re ( s ) < iclude imagiary axis. so his ROC is possible. For his ROC he iverse Laplace rasform is x () [ e u( ) e u( )] Hece (*) is correc aswer. SOL 6.79 For lef sided sequece we have au( ) Thus 5 u( ) or 5 u( ) z z z az where z < a 5z where z < 5 z z 5 where z < 5 Sice ROC is z < 5 ad i iclude ui circle, sysem is sable. Aleraive : h ( ) 5 u( ) Hz () / hz ( ) 5 / z /( 5z ) Le m, he m Hz () /( 5z ) /( 5 z) m m, 5 z < 5 z 5 z 5 z z 5 or z < 5 Page 46 Hece (B) is correc aswer. Published by: NODIA ad COMPANY ISBN:

55 SOL 6.8 s ( s ) s # s s # L ( * e ) u( ) s Here we have used propery ha covoluio i ime domai is muliplicaio i s domai LT X() s X () s x()* x() Hece (B) is correc aswer. SOL 6.8 We have h ( ) u ( ) Hz () / x ( ). z. z / /( z ) Hz () is coverge if / ( z ) < ad his is possible whe z <. Thus ROC is z < Hece (A) is correc aswer. or z > SOL 6.8 # We kow ha δ () x () x() δ() ad δ() Le x () cos( ), he x() Now δ() x () x() δ() d δ() d # # # Hece (A) is correc aswer. SOL 6.8 Le E be he eergy of f () ad E be he eergy of f( ), he E [()] f d ad E [( f )] d # # Subsiuig p we ge dp E # [()] fp [()] fp dp # Hece (B) is correc aswer. Published by: NODIA ad COMPANY ISBN: E Page 47

56 SOL 6.84 Sice h ()! for <, hus h () is o causal h () u() which is always ime ivaria, causal ad sable. u () h () is ime varia. + h4() e u() is ime varia. Hece (B) is correc aswer. SOL 6.85 h () f ()* g () We kow ha covoluio i ime domai is muliplicaio i s domai. L f ()* g () h () Hs () Fs ()# Gs () Thus Hs () s + s # + s + ( s+ )( s+ ) s + Hece (B) is correc aswer. SOL 6.86 Sice ormalized Gaussio fucio have Gaussio FT FT a Thus e a e f π π a Hece (B) is correc aswer. SOL 6.87 Page 48 Le x () ax() + bx() ay () ax() by () bx() Addig above boh equaio we have ay() + by() ax() + bx() [ ax( ) + bx( )] x() or ay() + by() y () Thus sysem is liear If ipu is delayed he we have yd ( d) x( ) If oupu is delayed he we have y ( ) ( ) x( ) which is o equal. Thus sysem is ime varyig. Hece (B) is correc aswer. Published by: NODIA ad COMPANY ISBN:

57 SOL 6.88 LS We have h () e H() s s LS ad x () e X() s s Now oupu is Ys () HsXs () () # s s s s Thus y () e e Hece (A) is correc aswer. SOL 6.89 We kow ha for a square wave he Fourier series coefficie ωτ si C A sq τ T ωτ Thus C sq \ If we iegrae square wave, riagular wave will be obaied, Hece C ri \ Hece (C) is correc aswer....(i) SOL 6.9 L u () u ( ) f () Fs () s [ e ] s L u () u ( ) g () Gs () s [ e ] s L f ()* g () FsGs () () s s [ e ][ e ] s or f ()* g () L s s s [ e e + e ] s e e e + s s s s s s s Takig iverse laplace rasform we have f ()* g () ( ) u( ) ( ) u( ) + ( ) u( ) The graph of opio (B) saisfy his equaio. Hece (B) is correc aswer. Published by: NODIA ad COMPANY ISBN: Page 49

58 SOL 6.9 Hece (A) is correc aswer. SOL 6.9 We have ft ( ) a T Takig z -rasform we ge T Fz () / a z /( a ) z Hece (A) is correc aswer. T / a T b z l z z a T SOL 6.9 If L [()] f Fs () Applyig ime shifig propery we ca wrie L[( f T)] e st F() s Hece (B) is correc aswer. SOL 6.94 Hece (A) is correc aswer. SOL 6.95 Hece (A) is correc aswer. SOL 6.96 Page 4 Give z rasform Cz () 4 z ( z ) 4 ( z ) Applyig fial value heorem lim f ( ) lim( z) f( z) " z " 4 z ( z ) lim( z ) F( z) lim( z ) z " z " 4 ( z ) 4 z ( z )( z) lim z " 4 ( z ) 4 4 z z ( z )( z) lim z " 4z ( z ) ( )( )( )( ) lim z z z+ z + z z " 4 ( z ) Published by: NODIA ad COMPANY ISBN:

59 lim z ( z+ )( z + ) z " 4 Hece (C) is correc aswer. SOL 6.97 We have Fs () ω + s ω lim f () fial value heorem saes ha: " lim f () lim sf() s " s " I mus be oed ha fial value heorem ca be applied oly if poles lies i ve half of s-plae. Here poles are o imagiary axis ( s, s! jω) so ca o apply fial value heorem. so lim f () cao be deermied. " Hece (A) is correc aswer. SOL 6.98 Trigoomeric Fourier series of a fucio x () is expressed as : x () A + [ A cos ω+ B si ω] / For eve fucio x, () B So x () / A + A cos ω Series will coai oly DC & cosie erms. Hece (D) is correc aswer. SOL 6.99 Give periodic sigal, x () *, T < T < < T The figure is as show below. Published by: NODIA ad COMPANY ISBN: Page 4

60 For x () fourier series expressio ca be wrie as / x () A + [ A cos ω+ B si ω] where dc erm A () T xd T/ xd () T T # T/ A T # T # T / # x() d x() d x() d T : + + D T/ T T + T + T T T Hece (C) is correc aswer. SOL 6. The ui impulse respose of a LTI sysem is u () Le h () u () Takig LT we have Hs () s If he sysem excied wih a ipu x () e a u(), a >, he respose Ys () XsHs () () Xs () L[ x ( )] ( s + a) so Ys () ( s + a) s a: s s + ad Takig iverse Laplace, he respose will be y () a e a Hece (B) is correc aswer. SOL 6. Page 4 We have x [ ] δ( k) / k Xz () xz [ ] / / ; / δ( k) z E k k Sice δ( k) defied oly for k so Xz () / z k z ( / z) ( z ) k Hece (B) is correc aswer. Published by: NODIA ad COMPANY ISBN:

61 SOL 6. Hece (B) is correc opio. SOL 6. x () Xf () by differeiaio propery; dx() F; d E jωx( ω) dx() or F; d E j πfx( f) Hece (B) is correc aswer. F SOL 6.4 F We have f () g( ω) by dualiy propery of fourier rasform we ca wrie F g () πf( ω) jω so F [ g ( )] ge () d πf( ω) # Hece (C) is correc aswer. SOL 6.5 Give fucio x () e α cos( α) L Now cos( α ) s s + α L If x () Xs () he s e x() L Xs ( s ) shifig i s-domai so α e cos( α) L ( s α) ( s α) + α Hece (B) is correc aswer. SOL 6.6 For a fucio x, () rigoomeric fourier series is : x () A + [ A cos ω+ Bsi ω] / where A () T # xd T Fudameal period T Published by: NODIA ad COMPANY ISBN: Page 4

62 A B x ()cosd ω T # T x ()sid ω T # T For a eve fucio x, () coefficie B for a odd fucio x, () A A so if x () is eve fucio is fourier series will o coai sie erms. Hece (C) is correc aswer. SOL 6.7 The cojugaio propery allows us o show if x () is real, he Xjω ( ) has cojugae symmery, ha is X( jω) X ) ( jω) [ x () real] Proof : replace ω by Xjω ( ) xe () ω he # X( jω) xe () X ) ) if x () real x () x() # jω jω d d jω ) jω ( jω) # xe () dg # x () e d ) ) jω he X ( jω) xe () d X( jω) Hece (C) is correc aswer. # *********** Page 44 Published by: NODIA ad COMPANY ISBN:

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