MATHA COLLEGE OF TECHNOLOGY

DEPARTMENT OF ECE-2ND YEAR IV TH SEM

MG UNIVERSITY PREVIOUS YEAR QUESTION PAPER FOR SIGNALS AND SYSTEMS

(10 YEARS)

TUTOR NAME:DIVYA R

MODULE 1

SOLUTION:REFER EC010403 SIGNALS AND SYSTEMS NOTES MODULE 1.PDF

GROUP -A

1.Define causality and bring out the important features of causal systems?

2.Explain periodic and aperiodic signals with examples?

3.Define and explain shift invariant and causality with examples for each?

4.Derive the expression for the output when two LTI systems are connected in parallel?

5.State and prove distributive property of convolution sum?

GROUP-B

6.Explain the following properties of LTI systems stability and causality?

7.Prove that the convolution representation of continuous time system is cumulative?

8.Differentiate between continuous and discrete time systems with examples?

9.Define and plot the following signals

i)unit impulse ii)unit step iii)unit ramp iv)unit exponential

10.Llist the properties of convolution sum?

GROUP -C

11.How do you define the invariance property of a continuous time system? give examples of a

time varying continuous and discrete time systems?

12.Explain commutative and associative properties of convolution sum?

13.How will you define invertibility property of a continuous time system give an example?

14.show that discrete LTI system is BIBO stable if its impulse response is absolutely summable?

1.Check whether the following systems are time invariant or not?

y(t)=sin x(t)

y(t)=t x(t) 2.determine which of the following systems are causal or non causal?

y(t)=x(t) cos(t+1)

y(t)=x(2t)

3.given y(t)=x(t) find whether it is memory less, causal, linear, stable and time invariant?

4.find the even and odd components of x(t)= jte

5.determine the following signals are periodic. if periodic find the period?

x(t)=v(t)+v(-t) where v(t)=sint u(t)

x(n)= )4/1sin()5/1cos( nn

6. Determine whether the following systems defined below is (i) casual (ii) linear (iii) dynamic (iv) time invariant (v) stable

y(t)=cosx(t)

7. Check whether each of the following systems defined below is (i) casual (ii) linear (iii) dynamic

(iv) time invariant (v) stable

y(n)+6y(n-2)=x(n)-x(n-2)

y(n)=x(n)-nx(n-1) 8. Determine whether the following systems defined below is (i) casual (ii) linear (iii) dynamic (iv) time invariant (v) stable

y(t)=x(2t) 9. Check whether the following signals are even or odd?

x(t)=2t

x(n)=sin (5n)

x(n)=sin (5n)cos(7n)

x(n)=sin (5n)-cos(7n)

10.determine the discrete signal x(n)=u(n) is power or energy or neither?

11.if x(n)=ane determine whether the signal is periodic or not. if yes find the period?

12. Check whether the following signals are causal or non causal?

y(t)=x(2t)

y(t)=

t

dttx )(

13.Test periodicity?

ate

jate

tcos sin 8t

tt 6sin3cos

t4sin2

14.Are the following signals periodic? if periodic find the period?

cos(t+ /4)

sin t+cos( t/4)

6+2 sin(6 t)=cos(4 t/5)

cos (n /2)-3 sin(n /4+ /3)

sin^2(3 n/5)

)13/( tje

15.Find the energy or power of the signal?

x(t)=t 0

1.convolute the following sequences?

2.convolute the following x(t)= te 2 u(t)

h(t)= te u(t)+ te2 u(-t) ?

3.convolute x(t) and h(t) where x(t) and h(t) are as shown in fig?

4.convolute the following?

5.h(n)= )()3/1(2)()2/1(3 1 nunu nn be the unit sample response of LTI system .if the input to this

sytem is a unit step ,find )(lim nyn

where y(n)=x(n)*h(n)

6.determine graphically the output of an lti system whose impluse response is

find the output of the system?

7.convolute the following continuous time signls x(t)= te 2 u(t) h(t)=u(t+2)?

8.by using convolution integral find out the response of the system to unit step input signal.implsue

response is h(t)=R/LLtRe / u(t)

9.convolute x(t)=cos( t)[(u(t+1)-u(t+3)] h(t)=u(t)

10convolute x(t)=t RCte / u(t) h(t)=1/RC RCte / u(t)

11determine the convolution of the two continous time functions?

x(t)=3 sin2t for all t

h(t)= t

e

12.find the convolution of the two discrete sequences?

x(n)=2^n u(-n-1)

h(n)=4^n u(-n-1)

13.find the output y(n) for n=1,2.....6 with h(n)= n2 u(n) and

)2(4)1(4)(2)( nnnnx

1.find the natural ,forced and total response of the system described by the difference equation

y(n)+.1y(n-1)-.3y(n-2)=2 u(n).given y(-1)=y(-2)=0

2.Determine the forced response of the system 5 dy(t)/dt +10y(t)=2x(t) for the input x(t)=2u(t)

3. find the total response of the system described by the difference equation y(n)-.5y(n-1)=5 cos (.4 n)

for n>=0 given y(-1)=4

4.determine the natural ,forced and total response d2y(t)/dt2+6dy(t)/dt+8y(t)=dx(t)/dt+2x(t) if y(0) =2;dy(0)/dt =

3 and x(t)= te 2 u(t)

5.Determine the natural response of the system 10dy(t)/dt+2y(t)=x(t) y(0)=2

6.find the output of the system when the input signal is x(t)=e-2t u(t) d2y(t)/dt2+5dy(t)/dt+6y(t)=6x(t)

7. 4.determine the natural ,forced and total response d2y(t)/dt2+5dy(t)/dt+4y(t)=dx(t)/dt if y(0) =0;dy(0)/dt = 1

and x(t)= te 2 u(t)

8.determine the natural ,forced and total response d2y(t)/dt2+6dy(t)/dt+8y(t)= 4te 3 u(t)

if y(0) =3;dy(0)/dt = 4

9..determine the natural ,forced and total response d2y(t)/dt2+4dy(t)/dt+4y(t)= 2te 2 u(t)

if y(0) =0;dy(0)/dt = 1

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MODULE II

SOLUTION:REFER EC010403 SIGNALS AND SYSTEMS NOTES MODULE 2.PDF

1. Find the Fourier transform of an impulse function x(t)= )(t .Also draw its spectrum?

2. State and explain briefly the time differentiation and convolution properties of continuous time Fourier

series?

3. State and explain briefly the time shifting and frequency shifting properties of continuous time Fourier

series?

4. state and explain any two properties of CTFT?

5. List any four properties of Fourier transform?

6. state and prove differentiation property of Fourier transform?

7. find the Fourier transform of the gate function shown below?

8. find the Fourier transform of the signal x(t)=t ate u(t).what is the restriction on 'a' for the Fourier

transform to exist?

9. use Parseval's theorem then calculate the energy in the signal x(t)=4 sinc 40t

10. find the Fourier transform of periodic signal x(t)= t0cos

11. find the Fourier transform of a rectangular pulse of period T?

12. explain the integration and differentiation properties of Fourier transform?

13. find the Fourier transform of x(t)= ta

e

for a>0

14. what is the condition for existence of Fourier transform?

15. state and prove frequency shift properties of Fourier transform?

16. using duality theorem find the Fourier transform of x(t)=1/1+t 2

17. find the Fourier transform of one sided exponential signal?

18. prove that )()( 00 jwXettxjwtFT

19. find the Fourier series of the following sweep voltage?

20. determine the time domain signal corresponding to the following Fourier transform

X(JW)=cos (w/2)+j sin(w/2) IwI

22. state and prove the convolution and modulation properties of Fourier transform?

23. Obtain the Fourier transform of x(t)=5[ u(t+3)+u(t+2)-u(t-2)-u(t-3)]?

24. State parseval's theorem? Verify the same for the energy signal x(t)=e at u(t) for t>0

25. Find the complex Fourier coefficient for the periodic waveform x(t) shown in fig. Draw the amplitude

and phase spectra?

26. Find the real and imaginary parts of the Fourier transform of the pulse x(t) shown in fig?

28.Find the Fourier series coefficients of the periodic signal

29.find the energy spectral density of the signal x(t)={e t2 t>=0}

30.derive parsevals theorem ,using the same find the energy of the signal )()( tuetx t

And also the energy in the frequency band 5. rad/sec?

31.find the frequency response of a continuous time LTI system represented by the impulse response h(t)=

tae

32.determine the complex exponential Fourier series expansion of the periodic signal?

34.find the frequency component of the signal?

x(t)={t for 0

)1(/1)(2 jj

47.find the Fourier transform of x(t) sketch and label the magnitude and phase spectra?

otherwise

tfor

tfor

tx

0

212

101

)(

48.explain the duality property of Fourier transform with examples? use the above property evaluate Fourier

transform of x(t)=1/1+jt

49.find the Fourier transform of the signal shown in figure below?

50.compute the Fourier transform of the signal?

51.find the Fourier transform of the signal?

{x(t)=1+cos (t) ItI

x(t)=1 for ItI

19. show that )()( 00

jXennxjDTFT

20. find the DTFT of x(n)=u(n)

21. x(n)=cos nw0 w 0 =2/5 evaluate DTFT?

22. find the DTFT of x(n)=

0

1

otherwise

NnNplot the magnitude and phase spectrum?

23. prove that convolution of two discrete signals is equal to the product of their Fourier transforms?

24. find the inverse DTFT of)65(

6)(

2

jj

j

eeeX

25. obtain the Fourier transform pair associated with the signal whose DTFT is )5.1(

1)(

j

j

eeX

26. find the inverse DTFT of )14/38/1(

4/53)(

2

jj

jj

ee

eeX

27. find the DTFT of x(n)=n a n u(n) for 1a

28. find the DTFT of the following sequences

x(n)= )2(3)1(6)( nnn

)2()3/1()( 2 nunx n

29.find the DTFT of x(n)=an

1a sketch magnitude and phase speturm?

30.finf the DTFT of x(n)=-a )1( nUn

31.find the DTFT of x(n)={1,2,3,2,1}

32.find the DTFT of periodic signal?

33.find the DTFT of y(n) given h(n)=(1/3) n u(n)

x(n)=(1/2) n u(n)

34.find the DTFT of signum function?

35.find the DTFT of rectangular pulse /plot its magnitude and phase spectrra?

36.find DTFT of x(n)=a n u(n)

37.find DTFT x(n)=(1/2) n u(n)plot its magnitude and phase spectrum?

38given DTFT of x(n) is )5.1(

1)(

j

j

eeX

find y1(n)=x(3n+1)

y2(n)=x(n) cos (.4n)

y3(n)=x(n-1)*x(n-2)

determine and plot the magnitude spectra?

39.determine the time domain signal to the Fourier transform of X(jw)=e w2 u(w)

40.Find inverse DTFT )16/56/1(

3/25)(

2

jj

jj

ee

eeX

41.find the natural response for y(n)+.1y(n-1)-.3y(n-2)=2u(n) the y(-1)&y(-2)=0

42.y(n)-.6y9n-1)=x(n)

x(n)=.4 0nn y(-1)=10 find the solution using described initial conditions?

43.find the impulse response and step response?

y(n)-3/4y(n-1)+1/8y(n-2)=x(n)

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MODULE IV

SOLUTION:REFER EC010403 SIGNALS AND SYSTEMS NOTES MODULE IV.PDF

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MODULE V

SOLUTION:REFER EC010403 SIGNALS AND SYSTEMS NOTES MODULE V.PDF

1.Find Z-transform ,ROC and sketch pole zero plot?

i) x(n)= )2(2 nun

ii) x(n)= )(])3(4)2(3[ nunn

iii) )1()( 1 nuanx n

iv)n

nx 2/1)(

v) )()( knnx

vi) )()( 0nnnx

vii) )()( nuanx n

viii) x(n)= 150])4/1()3/1[( nfornn

ix) nwnx 0sinh)(

x) )()4/sin()3/1()( nunx n

xi) )1()()2/1()( nununx n

xii) )1()2(/1)( nunnx n

xiii) )1()()( nubnx n

xiv) )()2/1()()3/2()( nununx nn

xv) )()3/1()()2/1()( nununx nn

xvi) )()3/1()1()2/1()( nununx nn

xvii) )1()3/1()()2/1()( nununx nn

xviii) )3()( nnnx

xix) )()4.(2)(3)(5)( nunuennx nn

2) )()()( 21 nxnxnx where )()3/1()(1 nunxn and )()5/1()(2 nunx

n find Z-transform?

3)Find Z-transform of ramp sequence?

4)find the inverse Z-transform of the following signal?

i) 1143

2)(

2

ZROC

zzzx

II) 34

4)(

2

zz

Zzx using partial fraction method?

iii) 12/1132

)(2

ZORZROCzz

Zzx

IV) 15.15

1)(

12

zzzx for using long division method?

v) 12/72/3

43)(

12

1

zz

zzx

vi) 112/32/1

51)(

12

1

ZROCzz

zzx

vii) 2112

)(12

1

ZROCzz

zzx

viii) 2/1)4/1()3/1(

2)(

2

2

ZROC

zz

Zzx

ix) 2/1)5.(

1)(

2

ZROC

z

Zzx

x) X(Z)=ze /1

xi) 2)1()2(

2)(

2

ZROC

zzzzx

xii) )25.(

5)(

2 zzzx

5)explain linearity property of z-transform?

6) explain time shift property of z-transform?

7)explain time reversal property of z-transform?

8) explain convolution property of z-transform?

9)properties of ROC?

10) Relation ship between DTFT and Z-transform?

11)illustrate the method of getting frequency response of a discrete LTI systems from its poles

and zeros with the help of examples?

12) find transfer function using z-transform y(n)+y(n-1)= )(n f0r 0n given y(-2)=1 y(-1)=0

13) find the transfer function and impluse response using Z-transform?

i) y(n)-1/2y(n-1)=2x(n-1)

ii) y(n)-1/4y(n-1)-3/8y(n-2)=-x(n)+2 x(n-1)

iii) y(n)=-1/2 y(n-1)+2 x(n)

13) find the transfer function and impluse response ,step response and check for stability using Z-

transform?

y(n)=.6y(n-1)-.08y(n-2)+x(n)

14.y(n)+y(n-1)-2y(n-2)=u(n-1)+2 u(n-2) find transfer function and impluse response with zero

intial condition?

15.y(n)= )()3/1(2 nun

x(n)=u(n) find h(n) using Z-transform?

16)x(n)= )()4/1(*)()2/1( nunun nn

17)x(n)={1,1,1,1} h(n)={1,1,1} find y(n) using convolution?

18. )2()()( nnnx )3()2(3)1()(2)( nnnnnh find y(n) using Z-

transform?

19.explain initial ,final and convolution theorem of Laplace transform?

20.Relation ship between CTFT and Laplace transform?

21.ROC of Laplace transform and its properties?

22.Relation ship between S-plane and Z-plane?

23.how steady state response is obtained from pole zero plot?

24.explain the significance of poles and zeros of the system transfer function on time response?

25,find the Laplace transform of sine wave

26. find the Laplace transform of cosine wave

27. find the Laplace transform of impulse function?

28.find the Laplace transform

i)x(t)= )(sin twtue at

ii)x(t)= )(cos twtue at

iii) x(t)= )()(2 tuetue tt

iv)x(t)= )()( 2 tuetue tt

v(x(t)=t cos 0w t u(t)

29.find the inverse laplace transform

i) 1)Re(1)2)(1)(1(

75)(

SROC

sss

ssx

ii) )23(

12)(

2

Ss

ssx

30. )(2)(2)(3)( ''' txtytyty using laplace transform determine transfer funcgion and impluse

respoinse and output response for the input x(t)=4 )(3 tue t

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