Kuestion Signals and Systems

Transcription

1 Kuestio Sigals ad Systems

2 Cotets Maual for Kuestio... Type : Delta Fuctio... 3 Type : Sigal Period... 4 Type 3: Sigal Represetatio... 5 Type 4: Sigal RMS Value... 9 Type 5: Eergy ad Power Sigals... Type 6: LTI System Properties... 3 Type 7: Covolutio... 6 Type 8: Covolutio by Graph... 8 Type 9: Cotiuous Time Fourier Series... Type : Cotiuous Time Fourier Trasform... 5 Type : Fourier Trasform by Graph... 6 Type : Properties of Fourier Trasform... 9 Type 3: Iitial Value i Fourier Trasform... 3 Type 4: Laplace Trasform Type 5: Iitial ad Fial Value Theorem Type 6: Magitude ad Phase Respose Type 7: Z-Trasform Type 8: Samplig... 4 Aswer Key Kreatryx. All Rights Reserved.

3 Maual for Kuestio Why Kuestio? It s very overwhelmig for a studet to eve thik about fiishig - questios per chapter whe the clock is tickig at the last momet. This is the reaso why Kuestio serves the purpose of beig the bare miimum set of questios to be solved from each chapter durig revisio. What is Kuestio? A set of 4 questios or less for each chapter coverig almost every type which has bee previously asked i GATE. Alog with the Solved examples to refer from, a studet ca try similar usolved questios to improve his/her problem solvig skills. Whe do I start usig Kuestio? It is recommeded to use Kuestio as soo as you feel cofidet i ay particular chapter. Although it will really help a studet if he/she will start makig use of Kuestio i the last moths before GATE Exam (November ed owards). How do I use Kuestio? Kuestio should be used as a tool to improve your speed ad accuracy chapter wise. It should be treated as a supplemet to our K-Notes ad should be attempted oce you are comfortable with the uderstadig ad basic problem solvig ability of the chapter. You should refer K-Notes Theory before solvig ay Type problems from Kuestio. 4 Kreatryx. All Rights Reserved.

4 Type : Delta Fuctio For Cocept refer to Sigals ad Systems K-Notes, Sigals Commo Mistake: Divide by the coefficiet of t i delta fuctio while itegratig. Sample Problem : What is the value of ( (t)cost (t )sit)dt (A)+cos4 (B)cos4 (C)+si4 (D)si4 Solutio: (C) is correct optio ( (t)cost (t )sit)dt cos() si(*) si4 Usolved Problems: Q. 6 (t t ) (t 3)dt 3 =? (A) 4.6 (B) 3.6 (C) 7 (D) Noe of the above Q. Fid the Fourier trasform of the sigal x(t) (t ) (t ) (A) j (B) j (C)cosω (D)Noe Q.3 The value ([t 3] (t ) 8cost (t.5))dt is (A)3.3 (B)3.56 (C)6.39 (D) 7.85 Q.4 The value 4 (t )sit)dt is (A)4si4 (B)si4 (C)si (D) 3

5 Type : Sigal Period For Cocept, refer to Sigals ad Systems K-Notes, Sigals Commo Mistake: If oe sigal period is ratioal ad other is irratioal, the sum of the two sigals ca ever be periodic. Sample Problem : The time period T of the sigal (A)π (B) 3 Solutio: (C) is correct optio t x(t) 3si 4cos(4t) 8si(t) (C) 4 (D) 3 is equal to Period of t 3si => T 4.5 Period of 4cos(4t) => T 4 Period of 8si(t) => T3 Now, Period of x t T L.C.M. T,T,T 3 L.C.M. 4,, 4 Usolved Problems: Q. The period of the sigal x(t)=8si(.8πt+π/4) is (A).4π sec (B).8π sec (C).5 sec (D).5 sec Q.. Let x() be a discrete time sigal, ad Let y() = x() & y x /, eve, odd 4

6 Cosider the followig statemets () If x() is periodic, the y() is periodic () If y() is periodic, the x() is periodic (3) If x() is periodic, the y() is periodic Which of the statemets are TRUE? (A),, 3 (B), 3 (C), (D), 3 Q.3 A discrete time sigal is give as x[] cos cos. The sigal is 8 8 (A) Periodic with period 6π (B) Periodic with period 6(π+) (C) Periodic with period 8 (D)Not Periodic Q.4 The period of the sigal 6e j 4 t j(3 t ) 3 8e is (A)4 (B)5 (C)6 (D)7 5t 6t t Q.5 The period of the sigal s(t) si 3cos 3si( 3 ) is? 7 (A)4π (B) π (C) 6π (D)Not periodic Type 3: Sigal Represetatio For Cocept, refer to Sigals ad Systems K-Notes, Sigals. Commo Mistake: Make sure that same argumet appears i fuctio as the uit step fuctio multiplied to it before takig ay Trasform. eg. The term with u(t-) should be (t-) as i sample problem. 5

7 Sample Problem 3: Which of the followig gives correct descriptio of the wave form. (A) u(t)+u(t-) (C) u(t)+u(t-) +(t-)u(t-) Solutio: (C) is correct optio (B) u(t)+(t-)u(t-) (D) u(t)+(t-)u(t-) Similarly i this questio For t< x(t)=u(t)-u(t-) For t< x(t)=[u(t-)-u(t-)] For t x3(t)=tu(t-)=(t-)u(t-)+u(t-) Now, x(t)= x(t)+x(t)+ x3(t)= u(t)+u(t-) +(t-)u(t-) Usolved Problems: Q. The fuctio f(t) show i the figure will have Laplace trasform as s s s s s (A) e e e e s s s (B) e e s e se s s s s s (C) (D) 6

8 Q. If a plot of sigal x(t) is as show i the figure The the plot of the sigal x(-t) will be Q.3 The odd part of the sequece x 6,4,, is 7

9 Q.4 Cosider two sigal x(t) ad y(t) show i figure ad there Laplace trasform pairs LT LT x(t) X(s) ad y(t) Y(s) If s s 4s X(s) 5 e 5e. The Y(s) will be? (A) s 4s 5 e 5e (B) s 4s 5 e 5e s s 4s (C) s5 e 5e (D) s Q.5 The iput output pair for a LTI system is give below. s 4s 5 e 5e Fid the output y[] if iput is as show below. 8

10 (A) y [] y [ ] y [ 3] (B) y [ ] y [ ] y [] (C) y [ ] y [ 3] y [ 5] (D) y [] y [ ] y [ ] Type 4: Sigal RMS Value Sample Problem 4: The rms value of the periodic waveform give i figure is (A) 6A B) 6 A (C) Solutio: (A) is correct optio Give I (t) T t, <t< T T 6, <t<t T T T t I dt dt 6 dt T T T T T T 44 36T I dt t dt T T T T 3 44 T I dt 8T T T T 8 =6+8=4 T Irms I dt 4 6A T 4 A 3 (D).5 A 9

11 Usolved Problems: Q. If curret of 6 si(t) 6 cos(3t ) 6 A passed through a true RMS 4 ammeter, the meter readig will be (A) 6 A (B) 6 A Q. The average value of the periodic sigal x(t) show i figure is (C) A (D) 6 A (A)5/6 (B) (C)5 (D)6 Q.3 Fid the R.M.S. value of the fuctio f(t) sit cos t across [,] (A) (C) si si (B) (D) cos cos Q.4 Fid the Average value of the fuctio f(t) t across [,3] (A) (B) 7 8 (C) 7 9 Type 5: Eergy ad Power Sigals For Cocept, refer to Sigals ad Systems K-Notes, Sigals Commo Mistake: (D) 4 9 A periodic sigal is always a Power Sigal ad a fiite sigal is always a Eergy Sigal. Sample Problem 5: The power i the sigal s(t) 8cos(t ) 4si(5t) is? (A)4 (B)4 (C)4 (D)8 Solutio: (A) is correct optio

12 s(t) 8cos(t ) 4si(5 t) s(t) 8si( t) 4si(5 t) Power lim s(t) dt T T T T T Power lim 8si(t) 4si(5t) dt T T T 8 4 Power 4 Usolved Problems: Q. The frequecy spectra of a x(t) is give below. The power of x(t) is (A) 4w (B).5w (C) 4w (D) 5w Q. Cosider a cotiuous time sigal x(t) (t ) (t ). The value of E for the sigal t x( )d? (A)J (B)4J (C).5J (D).5J

13 Q.3 The power cotaied i the first harmoics of periodic sigal show i figure below (A).383 W (B).33 W (C).67 W (D).38 W Q.4 Figure below shows the P.S.D. of a power sigal x(t). The the average power of the sigal is? (A) W (B) W (C)3 W (D)6 W Q.5 A ideal secod order low pass filter show below with cut off frequecy of rad/sec is supplied with x(t)=e -t u(t). Eergy at respose of the system (A) J (B) J (C) J 4 (D) J 8

14 Q.6 The discrete time sigal is give as f cos u u 6 3. The eergy of the sigal is? (A) (B) (C)3 (D)4 Type 6: LTI System Properties For Cocepts, refer to Sigals ad Systems K-Notes, LTI Systems Commo Tip: A LTI System by default is a Ivertible System. Sample Problem 6: The iput x(t) ad output y(t) of a system are related as t y(t) x( )cos(3 )d The system is. (A) time-ivariat ad stable (C) time-ivariat ad ot stable (B) stable ad ot time-ivariat (D) ot time-ivariat ad ot stable Solutio: (D) is correct optio X(t) y(t) = t y(t) x( )cos(3 )d X(t-t) o/p t = x( t )cos(3 )d, Let t tt x( )cos(3t 3 )d tt y(t t ) x( )cos(3 )d o / p y(t t ) The system is time-varyig or ot time-ivariat. For a bouded iput, x(t)=cos(3t)u(t) 3

15 t y(t) cos (3 )d t t = d cos(6 )d I I I is ubouded, I as t The system is ot stable. Usolved Problems: Q. The followig differece equatio represets iput output relatioship of a discrete system. y() y( ) 3y( ) 4x( ) The ature of the system is (A) Causal, Time Variat ad Ustable (B) Causal, Time Ivariat ad Ustable (C) Aticausal, Time Ivariat ad Ustable (D) No causal, Time variat, ad stable d y(t) d x(t) Q. The iput ad output relatioship of a system is give below tx(t) 6 dt dt (A) liear, time ivariat, causal (C) No-liear, time ivariat, causal (B) No-liear, time variat, o-causal (D) liear, time variat, o-causal Q.3 The impulse respose of a discrete LTI system is give by h() = - (.5) - u( 4). The system is (A) Causal & Stable (C) No causal & Stable (B) Causal & Ustable (D) No Causal & Ustable Q.4 The trasfer fuctio of a system is give by Hz (A) Causal ad stable (B) Causal, stable ad miimum phase (C) Miimum phase (D) Noe of the above z(3z ) The system is z z 4 4

16 Q.5 The system give below is (A) Liear ad Causal (B) No-Liear ad Causal (C) Liear ad No causal (D) No-Liear ad No Causal Q.6 Cosider a digital filter defied by the followig structure The rage of k for which the system is stable, is (A) k 3 (B) k 3 (C) k (D) 3 k 3 Q.7 The block diagram represetatio of a CT system i the figure below. The system is (A)BIBO Stable (B)BIBO Ustable (C)Margially Stable (D)Noe 5

17 Type 7: Covolutio For Cocept refer to Sigals ad Systems K-Notes, LTI Systems Commo Mistake: This is a legthy questio, take care of calculatios as this is the most probable error. Sample Problem 7: Give two cotiuous time sigals x(t)=e -t ad y(t)=e -t which exist for t>, the covolutio z(t)=x(t)*y(t) is (A) e -t - e -t (B) e -3t (C) e t (D) e -t + e -t Solutio: (A) is correct optio For x t e, t X(s) s t L.T. For y t e, t y(s) s z(t) x(t)* y(t), Z(s)=x(s) Y(s) t L.T. If Z(s)= (s )(s ) s s Takig iverse L.T. z(t)= (e -t - e -t )u(t) Usolved Problems: Q. The iput of a system x(t) = e 6t 6t 8t e e u(t) ad output y(t) u(t) The step Respose s(t) is 8t 8t (A) s( t) ( e ) u( t) (B) s( t) ( e ) u( t) 8 8 8t 8t (C) s( t) ( e ) u( t) (D) s( t) ( e ) u( t) 8 8 Q. Defie the area uder a cotiuous time sigal V(t) is Av the Ay=? V(t)dt if y(t)=x(t)*h(t), (A) Ax + Ah (B) AxAh (C) Ax - Ah (D) (Ax + Ah) 6

18 Q.3 Cosider the followig two sigals x() = (-) ; 4 h() = () ; 3 the covolutio is defied as y() = x()*h() the y() & y(3) respectively.. (A) 5, 3 (B) 3, 5 (C) -3, 5 (D) 3, -5 Q.4 Let x() = {, 5,, 4} ; h()={4,, 3} where x() is iput sigal of a discrete system ad h() is the impulse respose of the same the. The output y() of the system is (A) y() = {8,,,, 4, } (B) y() = {8,,, 4,, } (C) y() = {8,,, 3, 4, } (D) y() = {8,,, 4, 3, } Q.5 Sigals x(t) ad y(t) have the followig pole-zero diagrams The sigal g(t) ad h(t) defied as g(t)=x(t)e -3t ad h(t)=y(t)* e -t u(t). If g(t) ad h(t) are both absolutely itegrable, the (A) g(t) is left sided ad h(t) is right sided (B) g(t) is right sided ad h(t) is left sided (C) both g(t) ad h(t) are right sided (D) both g(t) ad h(t) are left sided Q.6 The impulse respose of a causal LTI system is give as h(t)=u(t)-u(t-6).the iput to this system show below The output of the system at t= sec is (A) (C)3 (B) (D)4 7

19 Q.7 The iput x() of a discrete system is give by x () 3, 4, 6, ad impulse respose h () 6,, 6. The umber of samples i y() ad y()are (A) 4, (B) 4, 8 (C)6, 8 (D) 6, -3 Type 8: Covolutio by Graph For Cocept, refer to Sigals ad Systems K-Notes, LTI Systems Commo Tip: Flip the fuctio by which the covolutio ca be easier to save time like uit step fuctio. Sample Problem 8: Fid the respose, whe the fuctio x(t) covolve with h(t) t< t t< (A) y(t) +5t t<4 4 4 t t< t t< (C) y(t) t+5t t<4 3 4 t t< t t< (B) y(t) +5t t<4 3 4 t (D)Noe Solutio: (B) is correct optio Covolutio of x(t) ad h(t) x t *h t x( )h(t )d Y(t)= 8

20 for t< y(t)= for t< t y(t) d t for t<4 t y(t) d 5d 5t 9

21 for 4 t 4 y(t) d 5d 3 Usolved Problems: Q. Two sigals x(t) ad y(t) are expressed as follows x(t) y(t) 4 t 8 t y(t) ca be expressed i terms of x(t) as (A) y(t) = x t (B) y(t) = x t (C) y(t) = x t 3 8 (D) y(t) = x t 3 8 Q. The covolutio of the sigals x(t) & h(t) show i fig. at t = 3 is h(t) x(t) t 3 t (A) (B) (C) (D) 4

22 Q.3 Give iput of system x(t) (t) (t ) (t ) ad impulse respose h(t) is show i figure below. The the output of the system is? Q.4 The graph show below represets a wave form obtaied by covolvig two rectagular waveform of duratio (A) 4 uits each (B) 4 ad uits respectively (C) 6 ad 3 uits respectively (D)6 ad uits respectively

23 Q.5 The iput x(t) to a liear time ivariat system ad the impulse respose h(t) of the system is show below The output of the system is zero everywhere except for the (A)<t<6 (B)<t<4 (C)<t<5 (D)<t<5 Type 9: Cotiuous Time Fourier Series For Cocept, refer to Sigals ad Systems K-Notes, Cotiuous Time Fourier Series Commo Mistake: Remember to divide by time period while calculatig Fourier Series Coefficiets. Sample Problem 9: The T.F.S. of the sigal x(t) is (A) A A sit si3t si5t A (B) sit si3t si5t A A (C) cost cos3t cos5t A A (D) sit cost si3t si3t

24 Solutio: (C) is correct optio For the give sigal, Fourier Series x(t) a a cos( t) b si( t). Period of the sigal T=π, ω= T A a x(t)dt Adt T T T 3 si( ) si A A a x(t)cos( t)dt Acos( t)dt Acos( t)dt si T 3 T A cos() b x(t)si( t)dt Asi( t)dt Asi( t)dt T 3 Hece x(t)= A A cost cos3t cos5t Usolved Problems: Q. Fourier series coefficiet of the wave form is (A) A ( ) k k (B) A ( ) k k (C) A ( ) k jk (D) A ( ) k jk 3

25 f(x) a a cos(x) b si(x). The Q. f(x), show i the figure is represeted by value of a is (A) (B)π/ (C) π (D)π Q.3 Let x(t) ad x(t) be cotiuous time periodic sigal with fudametal frequecy ω ad ω, Fourier series coefficiet C ad d respectively. Give that x(t)=x(t-)+x(-t) The Fourier coefficiet d will be C jc e j (A) j (B) j C jc e C C e (C) (D) j C C e Q.4 Determie the time sigal correspodig to the magitude ad phase spectra as show i the figure with ω=π (A) x(t) 4si4t cos3t 8 4 (B) x(t) 4sit cos3t 8 4 (C) x(t) 4cos4t cos3t 8 4 (D) x(t) 4cost si3t 8 4 4

26 Q.5 The expoetial Fourier series of a certai periodic fuctio is give as j3t jt jt j3t e e e e f(t) j j 3 j ( j) Fid the Fourier series of this fuctio (A) f(t) 3 4cos t cos 3t 4 (B) f(t) 3 4cos t 4 cos 3t 4 (C) f(t) 3 4 cos t 4cos 3t 4 (D) f(t) 3 4cos t 4 cos 3t 4 Type : Cotiuous Time Fourier Trasform For Cocept refer to Sigals ad Systems K-Notes, Fourier Trasform Sample Problem : Fid the Fourier trasform of the sigal (A) (B) a j Solutio: (D) is correct optio a a j at x(t) te u(t) : a> (C) (a j ) at jt at jt (aj )t X( ) te u(t)e dt te e dt te dt (aj )t (aj )t e e X( ) t (a j ) (a j ) X( ) (a j ) (a j ) (D) (a j ) Usolved Problems: Q. The Fourier trasform X(ω) of the sigal x(t) = sg(t) is (A) X ( ) / j (B) X ( ) / j (C) X( ) j / j (D) X ( ) ( ) / j 5

27 Q. Let the sigal x(t) have the fourier trasform X(ω). Cosider the sigal where td is arbitrary delay. The magitude of the fourier trasform of y(t) is? (A) x( ) (B) x( ) (C) x( ) (D) x( ) e d x(t t ) d y(t), j t d dt Q.3 If a sigal x(t)=+cos(πft)+cos(6 πft) is fourier trasformed, the umber of spectral lies i the Fourier trasform will be? (A)3 (B)4 (C)5 (D)6 FT jt Q.4 if x(t) X( ) the F.T. of y(t)=x(t-)e is j( ) (A) X( )e j( ) (B) X( )e (C) j( ) X( )e (D) j X( )e Q.5 Fourier trasform of (A) e jab F(a ) a (B) f( b) a is jab a e F(a ) (C) jab a e F(b ) (D) jab a e F(b ) Type : Fourier Trasform by Graph For Cocept refer to Sigals ad Systems K-Notes, Fourier Trasform. Commo Mistake: Remember to cosider the impulse fuctio while takig the derivative of the rectagular fuctio at the edges. Sample Problem : The Fourier trasform of the give sigal x(t) is cos si (A) 4j cos si (B) j (C) cos si 4j (D) cos si j Solutio: (B) is correct optio 6

28 Method equatio of x t tu t u t t u t u t t u t u t Takig Fourier Trasfer j e j e j e j e j j j X(j ) [ jsi ] cos cos si X(j ) j Method equatio of x t tu t u t Takig Fourier Trasform df.t. u t u t d si cos si X(j ) j j ( ) j( ) d d cos si X(j ) j( ) Usolved Problems: Q. The sigal x(t) is show as The the iverse Laplace trasform of X (s) is? 7

29 Q. For the sigal show below (A)Oly Fourier trasform exists (B)Oly Laplace trasform exists (C)Both Laplace ad Fourier trasforms exists (D)Neither Laplace or Fourier trasforms exists Q.3 Fid the Iverse Fourier trasform of X(ω) for the magitude ad phase spectra of X(ω) A cos t A si t (A) (C) A cos t t A si t t (B) (D) Q.4 Fid the iverse Fourier trasform of the spectra F(ω) depicted below 4 (A) sic (t)cos4t (B) sic (t)cos4t (C) sic (t)si4t 4 (D) sic (t)si4t 8

30 Q.5 Determie the Fourier trasform for the waveform show below (A) (B) (C) si(f) (3 e j ) f si(f) (3 e j ) f si(f) (3 e j ) f Type : Properties of Fourier Trasform For Cocept refer to Sigals ad Systems K-Notes, Fourier Trasform. Commo Mistake: Remember the duality property correctly for fuctios like rectagular ad triagular. Sample Problem : Let x(t) rectt where rect(x)= for - x ad zero other wise. si( x) The if sic(x), The fourier trasform of x(t)+x(-t) will be give by x (A) sic (C) sic cos( ) Solutio: (B) is correct optio For x(t) rectt (B) sic (D) sic cos( ) 9

31 si( ) si( f) X(f),X( ) sic(f) sic sa( ) f Note that Sic ad Sa fuctios are eve, i.e. Sic(λ)=Sic(-λ) Sa(λ)=Sa(-λ) Usig Time Reversal property of F.T. : FT x( t) X( f) or X( ) sic sic X( ) FT x(t) x( t) X( ) + X( ) X( ) Sic Usolved Problems: 4cost Q. The Fourier Trasform of. x(t) t (A) 3 4 (B) 4 (C) (D) Noe of the above 4 Q. The Fourier Trasform of. x(t) t (A) si (B) cos (C) 4 e (D) e Q.3 A phase shifter is defied by the frequecy respose j e H(j ) j e The the impulse respose h(t) will be (A) (B) (C) t t t (D) t 3

32 Q.4 Cosider a sigal g(t) x (t) x (t) where x (t)=sic5t ad x (t)=sict F.T. Also g(t) G( ) The amplitude A of the curve G(ω) is equal to -x. The x is (A) (B) (C)4 (D)5 4s 5s 8 Q.5 The iverse Laplace trasform of X(s) (s ) (s ). If the Fourier trasform of x(t) exists is? t t t (A) e u(t) te u(t) 3e u( t) t t t (B) e u( t) te u(t) 3e u(t) t t t (C) e u(t) te u(t) 3e u( t) (D) Noe Type 3: Iitial Value i Fourier Trasform For Cocept refer to Sigals ad Systems K-Notes, Fourier Trasform Sample Problem 3: For the sigal x(t) show i figure fid X() ad X( )d? (A)7,4π (B),6π (C)69,π (D)Noe Solutio: (A) is correct optio 3

33 jt x(t) X( )e d Put t= i above equatio x() X( )d Hece X( )d x() 4 j t ad X( ) x(t)e dt put ω= i above equatio X() x(t)dt area of the sigal 7 Usolved Problems: Q. For the sigal x(t) show below, Fid j X( ) d ad X( )e d (A) 7, (B) 8, 3 3 (C), (D) 8, 3 Q. The sequece x() represets output of system i discrete time domai. The value of X(π) would be? (A)3 (B)6 (C)9 (D) 3

34 Type 4: Laplace Trasform For Cocept refer to Sigals ad Systems K-Notes, Laplace Trasform Commo Mistake: While Calculatig Iverse Laplace Trasform, take care about Regio of Covergece of the Trasform like Laplace Trasform may be same for both left sided sigal ad Right Sided Sigal. Sample Problem 4: The Laplace trasform of (A) s e e 3 s s s (B) Solutio: (C) is correct optio (t t)u(t ) s e e 3 s s s is (C) s e e 3 s s s (D) Noe of the above Let x(t)=(t t)u(t ) (t ) u(t ) u(t ) Use the pairs ad properties of LT: u(t), u(t ) e s s LT LT tu(t), t u(t) 3 s s LT LT s (t ) u(t ) e s s s X(s) e e 3 s s LT s 3 Usolved Problems: Q. The O/p y(t) of a causal LTI system is related to the I/p x(t) as dy(t) y(t) x( )z(t )d x(t) dt. where x(t) = e -t U(t) + 3(t) the impulse respose is 7 7 t t e U(t) e U(t) (B) e U(t) e U(t) t t (A) (C) t t e U(t) e U(t) (D) (t) t e U(t) 33

35 Q. The output of the system show i figure below.if the iput x(t)= te -t u(t) is? (A) (B) (C) (D) t e t te t t e t t e t u(t) u(t) u(t) u(t) Q.3 The Laplace trasform of (sit)u(t t ) is (A) st e si t ta ( ) s s (B) s e s st sit ta ( ) (C) s e s st sit ta ( ) (D) s e s st sit ta ( ) Q.4 The Laplace trasform of a sigal x(t) is 9 X(s) ROC : -<Re(s)<. The sigal x(t) is? (s )(s ) t t t (A) t t t e u( t) e u( t) 3te u( t) (B) e u(t) e u( t) 3te u(t) t t t (C) t t t e u(t) e u(t) 3te u(t) (D) e u( t) e u(t) 3te u(t) s a Q.5 What is the iverse Laplace trasform of X(s) log s b t (A) x(t) coshbt coshat (B) x(t) sihat sihbt t (C) x(t) coshbt cosat (D) x(t) sihat sibt t t Type 5: Iitial ad Fial Value Theorem For Cocept, refer to Sigals ad Systems K-Notes, Laplace Trasform. 34

36 Commo Mistake: Check for Stability of a sigal before applyig Fial Value Theorem. Sample Problem 5: The uit impulse respose of a secod order uder-damped system startig from rest is give by 6t C(t).5e si8t, t. The steady state value of the uit step respose of the system is equal to (A) (B).5 (C).5 (D). Solutio: (D) is correct optio Impulse respose h(t)=.5e -6t si(8t), t si(8t)u(t) e 6t h(t) si(8t)u(t) L.T. 8 s 64 8 (s 6) 64 L.T. (s 6) 64 L.T. H(s) If the step respose is y(t) t y(t) h(t)dt, H(s) Y(s) (itegratig property of LT is used) s s (s 6) 64 yss lim y(t) limsy(s) (Fial value theorem of LT is used) t s Usolved Problems: L f(t) Q. (s ) the f() ad f( ) are? (s s 5) (A), (B), (C), (D).4, (s 3) Q. Iitial ad fial value of X(s) respectively are (s ) (A), (B), (C),.5 (D), 35

37 (s s ) Q.3 Determie the iitial ad fial values of the f(t), Give thatf(s) (s )(s s 4) (A),8 (B), (C),does t exist (D)Noe Q.4 The uit step respose of a system with the trasfer fuctio c()= ad c( )=, the the ratio a b (A)4 (B)5 (C)6 (D)5 is b(s a) X(s) (s b) is c(t).if 4 ( z ) Q.5 The z-trasform of sigal is X(z) z.what will be tha value of x[ ]? 4 ( z ) (A)/4 (B) (C) (D) Type 6: Magitude ad Phase Respose For Cocept refer to Sigals ad Systems K-Notes, Laplace Trasform. Sample Problem 6: I the system show i figure, the iput x(t) = sit. I the steady-state, the respose y(t) will be (A) si(t 45 ) (B) si(t 45 ) (C) Solutio: (A) is correct optio System respose H(s)= s Amplitude respose s H(j ) = j H(j )= j si(t 45 ) (D) si(t 45 ) Give iput frequecy = rad/sec S, H(j ) = 36

38 Phase respose h(j )=9 ta ( ) (j ) =9 ta () 45 h So the output of the system is y(t)= H(j ) x(t- h)= si(t 45 ) Usolved Problems: Q. I fig. the steady state output correspodig to the iput si t is 4 (A) 3 si(t 45 ) (B) 3 4 si(t 45 ) 3 4 (C) si(t 45 ) 3 4 (D) si(t 45 ) Q. Cosider a distortio less system H(ω) with magitude ad phase respose as show below. If a iput sigal xt cost si6 t is give to this system the output will be 37

39 (A) 4cost si6 t (B) 8cost si6 t 3 (C) 4cost si6t 6 3 (D) 8cost si6t Q.3 Let a sigal asi(ωt + φ) be applied to a stable liear time variat system. Let the correspodig steady state output be represeted as af(ωt + φ). The which of the followig statemet is true? (A) F is ot ecessarily a Sie or Cosie fuctio but must be periodic with ω = ω (B) F must be a Sie or Cosie fuctio with a = a (C) F must be a Sie fuctio with ω = ω ad φ = φ (D) F must be a Sie or Cosie fuctio with ω = ω Q.4 A causal LTI filter has the frequecy respose H(jω) show below. For the iput sigal jt x(t) e, output will be? (A) je jt jt (B) je (C) 4 je jt (D) 4 je jt s Q.5 Cosider a LTI system with system fuctio H(s). The steady state respose s 4s 4 of the system is give by (whe the excitatio is 8cost) (A) 4cos(t 45 ) (B) 8cos(t 45 ) (C) 8cos(t 35 ) (D) 38 8 cos(t 35 )

40 Type 7: Z-Trasform For Cocept refer to Sigals ad Systems K-Notes, Z-Trasform. Commo Mistake: Take care of ROC while calculatig iverse of Z-Trasform. Sample Problem 7: Cosider the D.E. y() y( ) x() ad x()= u().assumig the coditio of iitial 3 rest, the solutio for y(): is (A) 3 3 (C) Solutio: (B) is correct optio (B) 3 3 (D) Give D.E. y() y( ) x() 3 z X(z) u()z z Takig z Trasform ad Usig time shiftig property of z Trasform Y(z) y(z)z X(z) Y(z) z 3 3 z 3 Y(z) z 3z z 3z Takig iverse z Trasform Y(z)=

41 Usolved Problems: Q. The z trasform X(z) of the sigal x[] u( 3) is 4 (A) X(z) = 64 z 3 ; z 4z 4 (B) X(z) = 64 z 3 ; z 4z 4 (C) X(z) = 64z 4z ; z 4 (D) X(z) = 64z 4z 3 ; z 4 Q. Cosider three differet sigal x u x u u x3 u u Followig figure shows the three differet regio. Choose the correct for the ROC of sigal R R R3 (A) x[] x[] x3[] (B) x[] x3[] x[] (C) x[] x3[] x[] (D) x3[] x[] x[] 4

42 Q.3 H(z) is trasfer fuctio of discrete system ad has two poles at z = & H(z) is ratioal. ROC icludes z = 3 / 4, h() = & h( ) = 4. Impulse respose h() of the system is (A) h () = u() 4 u( ) (B) h () = u() 4 u( ) (C) h () = u() 4 u( ) (D) h () = u() 4 u( ) Q.4 Cosider the pole zero diagram of a LTI system show i the figure which correspods to trasfer fuctio H(z).. Match List I (The impulse respose) with List II (ROC which correspods to above diagram) ad choose the correct aswer usig the codes give below: {Give that H() = } List-I (Impulse Respose) List-II (ROC) P. [(- 4) + 6(3)]u[]. does ot exist Q. (- 4)u[] + (- 6)3u[- - ]. z > 3 R. (4)u[- - ] + (- 6)3u[- - ] 3. z < S. 4()u[- - ] + (- 6)3u[] 4. < z < 3 Codes : P Q R S (A) 4 3 (B) 3 4 (C) 4 3 (D) 4 3 4

43 Q.5 The iput-output relatioship of a system is give as y[] -.4y[ - ] = x[] where, x[] ad y[] are the iput ad output respectively. The zero state respose of the system for a iput x[] = (.4)u[] is (A).4 u (B).4 u.4 u (C) ( ).4 u (D) Q.6 The z -trasform of x[] = {, 4, 5, 7,, } 3 (A) z 4z 5 7z z,z 3 (B) z 4z 5 7z z,z z 3 (C) z 4 5 7z z, z 3 (D) z 4z 5 7z z, z z Q.7 The system diagram for the trasfer fuctioh(z), is show below z z The system diagram is a (A) Correct solutio (B) Not correct solutio (C) Correct ad uique solutio (D) Correct but ot uique solutio Type 8: Samplig For Cocept, refer to Sigals ad Systems K-Notes, Samplig Commo Mistake: For Bad Pass Sigals Nyquist Rate is differet from Low Pass Sigals. 4

44 Sample Problem 8: The frequecy spectrum of a sigal is show i the figure. If this is ideally sampled at itervals of ms, the the frequecy spectrum of the sampled sigal will be Solutio: (B) is correct optio Highest frequecy of the iput sigal, f h = KHz as show i the figure Samplig iterval, Ts= ms, fs= KHz => fs= fh, Therefore Aliasig or overlap of the adjacet spectra occurs i the sampled spectrum because fs< fh The Sampled spectrum : U*(j ) U*(f) f U(f f ) s s 43

45 The resultat spectrum U*(j ) is costat for all f as show i figure below Usolved Problems: Q. The Nyquist rate of the sigal x(t) 3cos (3 t) 5si (9 t) (A) 9 KHz (B) 4.5 KHz (C)6 KHz (D).5 KHz Q. A cotiuous sigal x(t) is obtaied at the O/p of a ideal LPF with cut off frequecy. If impulse trai samplig is Performed o x(t), which of the followig samplig Periods would guaratee that x(t) ca be recovered form its sampled versio usig a appropriate LPF? () T =.5m sec () T = m sec (3) T =.m sec (A), 3 (B),, 3 (C), 3 (D), Q.3 Let, m(t)=cos6πt + cos8πt, if it is sampled by rectagular pulse trai, as show i the followig figure. The spectral compoets i KHz preset i sampled sigal i frequecy rage KHz to 3KHz (A).3,.4,.6,.7 (B).6,.7 (C).3,.4 (D).,.4,.6,.8 Q.4 A sigal x(t) cos(5 t) is sampled with samplig iterval Ts=.5 sec ad passed through a ideal low-pass filter whose frequecy respose is show i the figure 44

46 The spectrum of output sigal will be? Q.5 Determie the Nyquist samplig rate for the sigal x(t) sic(5 t)sic( t) (A)5 (B) (C)5 (D)5 45

47 Aswer Key Type B C A D Type D B D C A Type 3 D A D B C Type 4 C A A D Type 5 B A A D C C Type 6 B B B D A A B Type 7 A B B C C A C Type 8 B C A D A Type 9 C A D C B Type A A C A B Type D B B B A Type D D A B A Type 3 B C Type 4 A A B D C Type 5 B D C B C Type 6 A C D B Type 7 A C C D C D D Type 8 B A B B D 46

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