B.Tech II Year I Semester Examinations, December - 2018

SIGNALS AND STOCHASTIC PROCESSES

(Electronics and Communication Engineering)

Time: 3 hours Max. Marks: 75

Note: This question paper contains two parts A and B.

Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 10 marks.

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PART - A (25 Marks)

1. 1. Define signal and system. (2 Marks)
   2. What is the significance of Hilbert transform? (3 Marks)
   3. Define cross correlation and auto correlation. (2 Marks)
   4. Write any three properties of Fourier transform. (3 Marks)
   5. What is an LTI system? Define it. (2 Marks)

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   6. Find the Laplace transform of tⁿ. (3 Marks)
   7. What is a random variable? (2 Marks)
   8. When is a random process said to be strictly stationary? (3 Marks)
   9. Define conditional probability. (2 Marks)
   10. Define probability density function. (3 Marks)

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PART - B (50 Marks)

(Answer any one full question from each unit)

UNIT - I

2. a) Explain orthogonality property of signals. (5 Marks)

   b) Determine whether the signal x(t) = cos(t) + sin(t) is periodic or not. If periodic determine its fundamental period. (5 Marks)

   OR

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3. a) Derive the expression for convolution of two signals. (5 Marks)

   b) Determine the convolution of the signals x(t) = e^-atu(t) and h(t) = e^-ßtu(t). (5 Marks)

UNIT - II

4. a) Explain about Dirichlet’s conditions for Fourier series. (5 Marks)

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   b) Find the Fourier transform of single symmetrical gate pulse of amplitude A, pulse width t. (5 Marks)

   OR

5. a) Explain about the properties of Fourier transforms. (5 Marks)

   b) Determine the Fourier transform of the signal x(t) = e^-atu(t). (5 Marks)

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UNIT - III

6. a) Distinguish between linear time invariant system and linear time variant system. (5 Marks)

   b) Find the transfer function, impulse response h(t) of an LTI system characterized by the differential equation (d²y(t)/dt²) + 5(dy(t)/dt) + 6y(t) = x(t). (5 Marks)

   OR

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7. a) Define and explain the properties of ROC of Laplace transform. (5 Marks)

   b) Find the Laplace transform of x(t) = e^-atsin?t u(t). (5 Marks)

UNIT - IV

8. a) Define probability density function and write its properties. (5 Marks)

   b) In a box, there are 100 resistors having the following values: 15 have 10O, 25 have 22O, 40 have 33O, 20 have 47O. If one resistor is picked up, what is the probability that it will be either 22O or 47O. (5 Marks)

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   OR

9. a) Explain about Gaussian random variable. (5 Marks)

   b) A random variable has the following probability function:

   x	-3	-2	-1	0	1	2	3
   P(x)	0.05	0.1	0.3	0.1	0.15	0.1	0.2

   Find i) E(x), ii) E(x²). (5 Marks)

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UNIT - V

10. a) Explain the concept of stationary and statistical independence. (5 Marks)

    b) Given the auto correlation function R_x(t) = e^-2|t|, find the power spectral density. (5 Marks)

    OR

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11. a) Derive the relation between input and output power spectral densities of an LTI system. (5 Marks)

    b) Find the auto correlation function corresponding to the power spectral density G(?) = 4/(1+(?²)). (5 Marks)

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