Model Question paper Programme Course Title Faculty Time : B.Tech (ECE) & B.Tech (ECM) Probability Theory and Random Processes : Semester : WIN 2020-21 Code : ECE2005 Max. Marks : : Dr. K. Mohanaprasad : Three Hours 100 Answer all questions Q.No. Sub. Sec. Question Description Marks Given that the function 1 [10] π(π₯ + π¦)2 − 2 < π₯ < 2 πππ − 3 < π¦ < 3 ππ,π = { } 0 πππ ππ€βπππ Find the constant b such that this is a valid joint density function and also determine the marginal density function ππ₯ (π₯) πππ ππ¦ (π¦). 2. Two statistically independent random variables X1 and X2 have the same probability density given by [5] 2π₯π 0 ≤ π₯π < π ππ (π₯) = { π2 πππ ππ€βπππ ππ π₯π 0 For i=1 and 2, where a > 0 is a constant. Find the exact density of the sum π = π1 + π2 and also compute mean and variance of W using central limit theorem. Page 1 of 6 3. [10] Two random variables X and Y are defined by πΜ = 0, πΜ = −1, Μ Μ Μ Μ π 2 = 2 , Μ πΜ Μ 2Μ = 4 πππ π ππ = 2 2 2 Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ , π Μ , π , π , ππ , −2. Two new random variable W and U are π = 2π + π, π = −π − 3π. Find π 2 ππ πππ π ππ . 4. Statistically independent random variable X and Y have moments π10 = 2, π20 = 14, π02 = [5] 12, πππ π11 = −6. Find the moment π22 . Page 2 of 6 5 6. 2 2 Gaussian random variables π1 πππ π2 , for which Μ Μ Μ π1 = 2, ππ1 = 9, Μ Μ Μ π2 = −1, ππ2 = 4 and πΆπ1,π2 = [5] −3, are transformed to new random variables π1 πππ π2 according to π1 = −π1 + π2 , π2 = −2π1 − 2 2 3π2 . Find πΜ 12 , Μ π22 , ππ1,π2 , πΜ π1 , πΜ π2 , πΆπ1,π2 . A Number of practical systems have “square-law” detectors that produce an output W(t) that is the [10] square of its input Y(t). Let the detector’s output be defined by W (t ) ο½ Y 2 (t ) ο½ X 2 (t ) Cos 2 (ο·0 t ο« ο± ) where ο·0 is a constant. X(t) is second-order stationary and ο± is a random variable independent of X(t) and uniform on (0, 2π). Find E[W(t)], Rww (t,t+τ) and check whether or not W(t) is wide-sense stationary. 7 Statistically independent, zero-mean random process X(t) and Y(t) have autocorrelation functions [5] ο΄ RXX (ο΄ ) ο½ e ο and RXX (ο΄ ) ο½ cos(2ο°ο΄) respectively. i. Find the auto correlation function of the sum W1(t) = X(t) +Y(t). ii. Find the auto correlation function of the difference W2(t) = X(t) -Y(t). iii. Find the cross correlation function of W1(t) and W2(t). Page 3 of 6 8 9 A stationary random sequence π₯(π) with mean µx=4 and auto covariance πΆπ (π) = [10] |π| ≤ 3 4 − |π|, { } is applied as input to a linear shift invariant system who impulse response 0, ππ‘βπππ€ππ π β(π) ππ β(π) = π’(π) − π’(π − 4) where π’(π) is a step sequence. The output of this system is another random sequence π¦(π). Determine the mean sequence ππ¦ , the cross covariance πΆππ (π) and the auto covariance πΆππ (π)ππ π¦(π). A causal LTI system, which is described by the difference equation [10] 1 1 π¦(π) = π¦(π − 1) + π₯(π) + π₯(π − 1) is driven by a zero mean WSS process with autocorrelation 2 3 π ππ (π) = 0.5|π| Determine the Power Spectral Density and the Cross Power Spectral Density. 10 A Random process X (t ) is applied to a network with impulse response h(t ) ο½ u (t ) t exp( οbt ) where b > 0 is a constant. The cross-correlation of X (t ) with the output Y(t) is known to have the [10] same form: RXY (ο΄ ) ο½ u(ο΄ ) ο΄ exp( οbο΄ ) . Find the autocorrelation of Y (t ) and the average power in Y (t ) . Page 4 of 6 11 Periodic samples of the autocorrelation function of white noise N(t) with period Ts are defined by 2 π=0 π ππ (πππ ) = {ππ } 0 π≠0 A random sequence Y[n] is formed by adding the white noise sequence N[n] to a one-unit delayed white noise sequence according π[π] = π[π] + π1 π[π − 1] Where b1 is a real constant. Find the autocorrelation of Y[n] and the power density spectrum of white noise and Y[n]. 12 An Amplifier has three stages for which Te1=150 K (First stage), [10] Te2 = 350K and Te3 = 600K (Output stage). Available power gain of the first stage is 10 and overall input effective noise temperature is 190K. a) What is the available power gain of the second stage? b) What is the cascade’s standard spot noise figure? c) What is the cascade’s operating spot noise figure when used with a source of noise temperature Ts= 50K? [10] Page 5 of 6 Total marks [100] οοο Page 6 of 6