SIGNALS & SYSTEMS

QUESTION BANK – 3

1. a) Explain how Fourier Transform can be derived from Fourier series.b) Distinguish between Fourier series and Fourier transform.c) Distinguish between the expression form of Fourier series and Fourier transform. What is the nature of the “transform pair” in the above two cases?

2. State the conditions for the existence of Fourier transform of a signal.3. Explain Fourier transform of an arbitrary signal.4. Determine the Fourier transform of the following time function:

x ( t )= e−3 t [u ( t+2)− u ( t−3)] .5. Obtain the Fourier transform of the following functions:

(i) Impulse function f(t)(ii) DC signal(iii) Unit step function

6. Evaluate the Fourier transform of the function:

7. An AM signal is given by f ( t )= 15 sin (2π 106 t ) + [5 cos 2 π 103 t + 3 sin 2 π102 t ] sin 2 π106 t . find the Fourier transform and its spectrum.

8. Determine the inverse Fourier transform of the spectrum shown below:

9. Find the Fourier transform of the following signal:

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10. Explain the concept of Fourier transform for periodic signals.11. Find out the Fourier transform of the periodic pulse train shown in fig.

12. Find the Fourier transform of the following functions:(i) A single symmetrical triangular pulse(ii) A single symmetrical gate pulse(iii) A single cosine wave at t = 0.

13. Find the Fourier transform of cos0t and draw the spectral density function.14. If the waveform V(t) has the Fourier transform V(f), then show that the waveform delayed

by time t0 i.e., V(t – t0) has the transform of V ( f ) e− jωt0.

15. Find the Fourier transform for the signals, ( i) x ( t )= δ( t−2 )( ii ) x ( t )= 25 sin c (10( t−2 )).

16. Power signals will have Fourier transforms and energy signals will have Fourier series in the frequency domain. Justify this statement.

17. Find the Fourier transform for the following function

18. Find the total area under the function g( t ) = 100 sin c (( t−8 )/30) .19. Compute the Fourier transform for the signals given below:

( i) 3 cos (10 t ) + 4 sin(10 t )( ii ) 2 e(−1+ j 2 π )t u( t ) + 2 e(−1− j2 π ) t u ( t )

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20. Find the Fourier transform of the following waveforms shown in fig.

21. Find the Fourier transform of the signal x ( t )= 2

1+t2

22. Find the Fourier transform of the signal shown in fig.

23. For the following signal, find the power, rms value and sketch the PSD:

A cos 40 t + B sin 60 t24. State and explain Homogeneity, Superposition and Linearity properties.25. Derive the expression for energy density spectrum function of energy signal f(t) from

fundamentals and interpret why it is called energy density spectrum.26. State and prove Rayleigh Energy theorem (Parseval’s theorem for non-periodic functions).27. State and prove the Multiplication in time domain property of Fourier transform.28. State and prove Symmetry (Duality) property, time scaling property of the Fourier

transform.29. State and prove the convolution property in relation to Fourier transform.30. Explain in detail with suitable examples any two properties of Fourier transform.31. Determine the Fourier transform of a trapezoidal function and triangular RF pulse f(t)

shown in fig. Draw its spectrum.

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32. Using the integration property, find the Fourier transform for the function,

g( t ) = { 1 |t|<12−|t| 1<|t|<20 elsewhere

33. If f(t) F(), show that

dn tdt n

↔ ( jω)n F (ω )

34. Find the Fourier transform of the Signum function and plot its amplitude and phase spectrum.

35. Find the Fourier transform for the following functions shown in fig.

36. What is Hilbert space? Explain its properties in detail.37. Differentiate the Hilbert space of energy & power signals.

38. Find the Fourier transform of the function f ( t )= 5 sin2 3 t .

39. Determine the Fourier transform of two sided exponential pulse x ( t )= e−|t|.

40. Find the Fourier transforms of an even function xe(t) and odd function x0(t).

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41. Find the Fourier transform of sin (8 t+0 .1π )42. Obtain the Fourier transform of the following:

( i) x ( t )= A sin (2 πf C t ) u( t )( ii ) x ( t )= f ( t ) cos (2πf C t + φ )

43. Find the Fourier transform of f ( t )= {cos πt −1/2≤ t ≤ 1/2

0 otherwise

44. A waveform V(t) has a Fourier transform which extends over the range from –fm to +fm

which extends over the range from -2fm to +2fm. Show that V2(t) has a Fourier transform.45. Find and sketch the Inverse Fourier transform of the waveform shown in fig.

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