Random Signal Analysis - May 2014

Electronics & Telecomm. (Semester 5)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks. 1 (a) Explain any two properties of cross correlation function(5 marks) 1 (b) State and prove any two properties of Probability Distribution Function(5 marks) 1 (c) Define Strict Sense Stationary and Wide Sense Stationary Process.(5 marks) 1 (d) State and explain joint and conditional Probability of events(5 marks) 2 (a) Box 1 contains 5 white balls and 6 black balls. Box 2 contains 6 white balls and 4 black balls. A box is selected at random and then a ball is chosen at random from the selected box.
(i) What is the probability that the ball chosen will be a white ball?
(ii) Given that the ball chosen is white, what is the probability that it came from Box 1?(8 marks) 2 (b) The joint Probability density function of (x,y) is given by
f_xy(x,y)=Ke^-(x+y); 0<x<y<? <br=""> Find : K
(i) Marginal densities of x and y
(ii) Are x and y independent? </x<y<?>(12 marks) 3 (a) If X and Y are two independent random variables and if Z=X+Y, then prove that the probability density function of Z is given by convolution of their individual densities.(10 marks) 3 (b) Find the characteristics function of Binomial Distribution and Poisson Distribution.(10 marks) 4 (a) Define Central Limit Theorem and give its significance(5 marks) 4 (b) Describe sequence of random variables(5 marks) 4 (c) State and prove Chapman-Kolmogorov equation. (10 marks) 5 (a) Find the autocorrelation function and power spectral density of the random process x(t)=a cos(bt+Y) where a,b and constant and Y is random variable uniformly distributed over (-π, π)(10 marks) 5 (b) Show that the random process given by
x(t)=A cos(w₀t+θ)
Where A and w₀ are constant and θ is uniformly distributed over (0, 2π) is wide sense stationary(10 marks) 6 (a) Explain power spectral density function. State its important properties and prove any one of the property. (10 marks) 6 (b) Prove that if input to LTI system is WSS then the output is also WSS(10 marks) 7 (a) Prove that the Poisson process in Markov Process(5 marks) 7 (b) The transmission matrix of Markov chain with three state 0,1,2 is
$$given\ by\ \ P=\begin{array}{cc}& \\&\end{array}\begin{array}{cc}\ & \begin{array}{c}0 & \ \ \ \ \ \ 1 & \ \ \ \ \ \ 2\end{array} \\\begin{array}{ccc}0 \\1 \\2\end{array} & \left[\begin{array}{ccc}0.75 & 0.25 & 0 \\0.25 & 0.5 & 0.25 \\0 & 0.75 & 0.25\end{array}\right]\end{array}$$
and the initial state distribution is
P(x₀=i)= 1/3, i=0,1,2.
Find : (i) P[x₂=2]
(ii) P[x₃=1, x₂=2, x₁=1, x₀=2]
(10 marks) 7 (c) Define Markov Chain with an example and application(5 marks)