## 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(w_{0}t+θ)

Where A and w_{0} 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(x_{0}=i)= 1/3, i=0,1,2.

Find : (i) P[x_{2}=2]

(ii) P[x_{3}=1, x_{2}=2, x_{1}=1, x_{0}=2]

(10 marks)
**7 (c) ** Define Markov Chain with an example and application(5 marks)