Question Paper: Random Signal Analysis Question Paper - December 2016 - Electronics and Telecom Engineering (Semester 5) - Mumbai University (MU)
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Random Signal Analysis - December 2016

MU Electronics and Telecom Engineering (Semester 5)

Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary
1(b) State and explain Chebyshev's inequality. 5 marks

1(c) State Central Limit theorem and gives its significance. 5 marks

1(d) state and explain Baye's theorem. 5 marks

2(a) A two dimensional Random varible has the following pdf.
fxy(x, y)=kxye-(x2+y2),x>0, y>0
Find
i) Value of constant K.
ii) Marginal density of X and Y.
iii) Conditional densities of X and Y.
iv) Check for independence of X and Y.
5 marks

2(b) In a communication system, a zero in transmitted with probability 0.3 and a one is transmitted with probability 0.7. Due to noise in the channel, a zero is received as one with probability 0.2. Similarly, a one is received as zero with probability 0.4. Now,
i) What is the probability than a one is received?
ii) It is observed that a one is received. What is the probability that zero was transmitted
iii) What is the probability that an error is committed?
5 marks

3(a) If the joint pdf of (X, Y)is given as, f xy (x, y)= e-(x+y) x>0, y>0 Find the probability density function of (U, V), where U=xx+yand  V−X+Y.U=xx+yand  V−X+Y. U=\frac{x}{x+y}\text{and} \ \ V-X+Y. / Are U and V independent? 5 marks

3(b) Define Moment Generating function of a Random variable. If X is a RV discrete or Continuous, then show that its nth raw moment is given as, <mi>E</mi><mo stretchy="false">(</mo><msup><mi>X</mi><mi>n</mi></msup><mo stretchy="false">)</mo><mfrac><mrow><msup><mi>d</mi><mi>n</mi></msup><mi>M</mi><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><msup><mi>t</mi><mi>n</mi></msup></mrow></mfrac><mtext> </mtext><mtext>at</mtext><mtext> </mtext><mtext> </mtext><mi>t</mi><mo>=</mo><mn>0.</mn>[/itex]" role="presentation" style="font-size: 125%; text-align: center; position: relative;">E(Xn)dnMx(t)dtn at  t=0.<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">(</mo><msup><mi>X</mi><mi>n</mi></msup><mo stretchy="false">)</mo><mfrac><mrow><msup><mi>d</mi><mi>n</mi></msup><mi>M</mi><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><msup><mi>t</mi><mi>n</mi></msup></mrow></mfrac><mtext> </mtext><mtext>at</mtext><mtext> </mtext><mtext> </mtext><mi>t</mi><mo>=</mo><mn>0.</mn>[/itex]<script type="math/tex; mode=display" id="MathJax-Element-2">E(X^n)\frac{d^nMx(t)}{dt^n} \ \text{at} \ \ t=0.</script> 5 marks

4(a) Let X1,X2,X3,....... be sequence of Random variables. Define i) Convergence almost everywhere
ii) Convergence in probability
iii) Convergence in distribution
iv) Convergence in mean square sense for the above sequence of Random variable X.
5 marks

4(b) Prove that if input LTI system is WSS process, then its output is also a WSS processs. 5 marks

5(a) A Random process is given byX(t)=Acos(ωt+θ),X(t)=Acos⁡(ωt+θ), X(t)= A\cos \left ( \omega{t} +\theta \right ), / where A and ω are constants and θ is a Random variable that is Uniformly distributed in the interval (0;2π). Show that X (t) is WSS process and it is Correlation ergodic. 5 marks

5(b) Explain Power spectral density and prove any two of its properties. The power spectrum of a WSS process is given by,<mi>S</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>10</mn><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mn>25</mn></mrow><mrow><mrow><mo>(</mo><mrow><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn></mrow><mo>)</mo></mrow></mrow></mfrac>[/itex]" role="presentation" style="font-size: 125%; text-align: center; position: relative;">S(ω)=10ω2+25(ω2+4)(ω2+9)<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>S</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>10</mn><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mn>25</mn></mrow><mrow><mrow><mo>(</mo><mrow><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn></mrow><mo>)</mo></mrow></mrow></mfrac>[/itex]<script type="math/tex; mode=display" id="MathJax-Element-4"> S(\omega )=\frac{10\omega ^2+25}{\left ( \omega ^2+4 \right )\left ( \omega ^2+9 \right )}</script> Find its autocorrelation function. 5 marks

6(a) State and prove Chapman-Kolmogorov equation. 5 marks

6(b) The transition probability matrix of a Markov chain {Xn} n=1,2.......,having three states 1,2 and 3 isP=123123⎡⎢⎣0.10.50.40.60.20.20.30.40.3⎤⎥⎦P=123123[0.10.50.40.60.20.20.30.40.3] P=\begin{matrix} 1\\ 2\\ 3 \end{matrix}\begin{matrix} \begin{matrix} 1 &2 &3 \end{matrix}\\ \begin{bmatrix} 0.1 & 0.5 & 0.4\\ 0.6& 0.2 &0.2 \\ 0.3& 0.4& 0.3 \end{bmatrix} \end{matrix} / The initial probability distribution is P(0)=(0.7, 0.2, 0.1)
Find i) P (X2)=3
ii) P(X3=2, X2=3, X1=3, X0=2)
5 marks