PTU B.Tech Robotics Engineering 3rd Semester May 2019 63001 MATHEMATICS III Question Papers

Code: 13A05305

R13

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

B. Tech II Year I Semester Examinations, November/December - 2016

SIGNALS AND SYSTEMS

Time: 3 Hours Max. Marks: 75

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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.

PART - A (25 Marks)

1. 1. Define Signal and System. Give examples. [2M]
   2. Write short notes on unit step function. [3M]

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   3. Define Fourier series. [2M]
   4. Determine the Fourier transform of the signal e^-atu(t). [3M]
   5. Define an LTI system. [2M]
   6. What is an ideal filter? [3M]
   7. Define Z-transform. [2M]

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   8. List the properties of ROC for Z-transform. [3M]
   9. Define state. [2M]
   10. Write the properties of state transition matrix. [3M]

PART - B (50 Marks)

(Answer any one full question from each unit)

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UNIT - I

1. 2.a) Determine whether the signals are periodic or not.

   1. x(t) = cos(t) + sin(v2t)
   2. x(n) = cos(pn/2) + cos(pn/4) [5+5]

   b) Explain orthogonality property between two complex functions f₁(t) and f₂(t) over a time interval t₁ = t = t₂.

   OR

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2. 3.a) Approximate the function f(t) = t, 0 < t < 1 by a constant over the interval (0, 1). Using the mean square error criterion.

   b) Discuss the classification of continuous time systems.

UNIT - II

1. 4.a) Find the Fourier series representation of an impulse train given by

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   x(t) = ?_n=-8⁸ d(t-nT)

   b) State and prove differentiation in time and frequency domain properties of Fourier transform.

   OR

2. 5.a) Find the Fourier transform of x(t) = e^-a|t|, a > 0.

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   b) Explain the concept of Hilbert transform. [5+5]

UNIT - III

1. 6.a) What is distortion less transmission? What are the conditions for distortion less transmission?

   b) Determine the transfer function and impulse response for the LTI system described by

   y''(t) + 5y'(t) + 6y(t) = x'(t) + 8x(t) [5+5]

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   OR

2. 7.a) Explain the relationship between bandwidth and rise time.

   b) Find and sketch the output of an LTI system whose impulse response is h(t) = u(t) - u(t-2) for the input x(t) = e^-tu(t). [5+5]

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UNIT - IV

1. 8.a) Find the Z-transform of x(n) = n²u(n).

   b) State and prove initial and final value theorems. [5+5]

   OR

2. 9.a) Determine the inverse Z-transform of X(z) = z/(3z²-4z+1) , ROC |z|>1.

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   b) Check whether the system function H(z) = (2z²-0.8z+5) / (z²-0.8z+0.64) is stable or not. [5+5]

UNIT - V

1. 10.a) Obtain the state model of the system governed by the differential equation

   d²y(t)/dt² + 3dy(t)/dt + 2y(t) = dx(t)/dt + 5x(t)

   b) List the properties of state transition matrix. [5+5]

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   OR

2. 11.a) Explain the concepts of controllability and observability.

   b) The state equations of a network are given by

   X'(t) = [
   

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       0 1
       -2 -3 ] X(t) + [
       0
       2 ] u(t)
   Y(t) = [1 0] X(t)

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   Determine the transfer function of the network.

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