Discrete Time Signal & System - May 2012

Electronics Engineering (Semester 6)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks. 1 (a) Find the Z-transform and hence DFT of:-
x(n) = (0.25)ⁿu(n+4)(5 marks) 1 (b) Find the magnitude and phase response of the system described by the difference equation:-
y(n) = x(n) + [(1/3) x(n-1)] + [(1/6) x(n-2)](5 marks) 1 (c) Derive Parseval's Energy Theorem. Also state the significance of the same.(5 marks) 1 (d) Calculate the speed improvement factor in calculating 256 point DFT of sequence using direct computation and using FFT algorithm.(5 marks) 2 (a) Find Energy and power of the following signals:-
(i) x(n) = (1/3)ⁿ for n≥0 = 3ⁿ for n<0
(ii) x(n) = 3 cos(4πn + θ)(8 marks) 2 (b) Test the following systems for linearity and time invariance:-
(i) y(n) = a^x(n)
(ii) y(n) = 2x(n) + 1/[x(n-1)] (6 marks) 2 (c) Determine the transfer function, impulse response and step response for the system given by:-
y(n) = y(n-1) + x(n) + x(n-1)(6 marks) 3 (a) Determine the inverse ZT for
X(z) = z/[(3z² - 4z + 1)
ROC |z|>1 (8 marks) 3 (b) Obtain Cascade and Parallel realization of the second order DT linear system defined by:-
y(n) - 0.75 y(n-1)+ 0.125 y(n-2) = x(n) + 0.3 x(n-1) (12 marks) 4 (a) Determine the circular convolution of the two sequences x₁(n) and x₂(n) if-
x₁(n) = δ(n) + δ(n-1) - δ(n-2) - δ(n-3) and
x₂(n) = δ(n) - δ(n-2) + δ(n-4)(10 marks) 4 (b) For the pole-zero plot of a system shown below:-

(i) Find the equation of magnitude response.
(ii) Plot the magnitude response.
(iii) Identify the type of system based on pass band. (10 marks) 5 (a) Given x(n)=2ⁿ and N=8, find X(k) using DITFFT algorithm.(10 marks) 5 (b) For x(n) = {1+5j, 2+6j, 3+7j, 4+8j}, find DFT X(k). Using the result above and not otherwise, find DFT of the following sequences:-
x₁(n) = {1,2,3,4} and
x₂(n) = {5,6,7,8}(10 marks) 6 (a) Explain the overlap-add and overlap-save method for filtering of long data sequences.(10 marks) 6 (b) Draw the functional block diagram of TMS320C5X processor and explain various functional units.(10 marks) 7 (a) Using DFT/IDFT, find circular convolution of
x₁(n) = {1,1,2,2} and x₂(n) = {1,2,3,4}(10 marks) 7 (b) Classify the following systems as FIR/IIR, Min/Max/Mixed Phase and specify the passband from the pole-zero plot shown below:-
(10 marks)