2Problem 1: Two jointly zero–mean normal random variables X1 and X2 have a joint probability density function (PDF): fX1,X2(x1, x2) = 2 ⇡ p 7 e 8 7(x21+ 3 2x1x2+x 2 2) (a) Determine the covariance matrix: KX1X2 = 24Var(X1) Cov(X1,X2) Cov(X2,X1) Var(X2) 35 (b) Find the matrix A such that a new set of variables Y1 and Y2:24Y1 Y2 35 = A 24X1 X2 35 are independent, each having zero mean and unit variance. You may use MATLAB to do the factorization of the covariance matrix (refer to the svd function). MATLAB (c) Consider simulating the linear minimum-square error (MSE) prediction of X2 given X1. (i) Generate 10000 values of X1 and X2 by applying the inverse transformation A 1. Plot them in one plot using the subplot function. (ii) Estimate the means, variances, and correlation coefficient ⇢ using the generated X1 and X2. (iii) Use the estimates to calculate the parameters ↵ and of the estimator: Xˆ2 = ↵+ X1. Compare them to the actual values. (iv) With the determined parameters ↵ and use the values of X1(n) to find estimates Xˆ2(n) for n = 0, · · · , 9999. Plot the generated X2(k) and the estimated Xˆ2(k) for k = 0, · · · , 99 in a second figure (you will put 3 figures together —this and the two that follow— by using subplot, use red for X2(k)). (v) Compute the estimation error “(n) = X2(n) Xˆ2(n) for n = 0, · · · , 9999. Plot ✏(k) for k = 0, · · · , 99 in the same plot as above. (vi) Plot a histogram of the estimation error “(n) in the above plot. (vii) Compute the theoretical minimum MSE “min and compare it with the estimate of E[“(n) E(“(n))]2 obtained using MATLAB. MATLAB (d) Change the correlation coefficient ⇢ to 0.95 and leave the other parameters the same, use your script and functions to simulate the linear MSE estimation of X2 from X1 and compare these results with the ones obtained before. 3Problem 2: A wide sense stationary (w.s.s.) random process X(n) with zero mean is character- ized by the following difference equation: X(n) = aX(n 1) + U(n) 1 < n <1 where |a| < 1, and U(n) is a white Gaussian noise (WGN). The WGN random process U(n) has zero mean, variance 2U , and its samples are all independent with Gaussian PDF. (a) Express X(n) only in terms of the samples of the random process U(n). (b) Is X(n) a Markov process? Can you tell whether or not X(n) is Gaussian? Justify your answer. (c) Prove the following expression: RX(k) = a |k|RX(0) 1 < k <1 (d) Show that the average power of X(n) can be computed as: RX(0) = 2U 1 a2 Show each step of your work. MATLAB (e) Assume 2U = 1 a2. Write a MATLAB function to generate the following plots: (i) Consider a = 0.25. Generate a plot with the realization of the random process X(n) for n = 0, · · · , 50. In the same figure, use subplot to display the autocorrelation function RX(k) for k = 10, 9, · · · , 9, 10. (ii) Consider a = 0.98 and redo the figure from part (i) with the realization of X(n) and the autocorrelation function RX(k). Comment on the differences between the two cases. Problem 3: A wide sense stationary stochastic process X(t) with mean E [X(t)] = 5 and power spectral density SX(f) = 1/(1 + ⇡2f 2) is the input to a filter with frequency response H(f) = 1/(1 + j2⇡f) such that X(t) ⇤ h(t) = Y (t), where ⇤ is the convolution operator and h(t) is the impulse response of the filter. Find the following: (a) The mean E [Y (t)] of the process Y (t). (b) The power spectral density SY (f) of the process Y (t). (c) The autocorrelation RY (⌧) of the process Y (t). (d) The average power E{Y 2(t)} of the process Y (t). 4Problem 4 (Bonus): Let X and Y be two independent exponential random variables with com- mon parameter . Define the random variables U and V using the following linear transformation: U = X + Y V = X Y (a) Find the joint probability density function fUV (u, v). Specify the ranges for U and V . (b) Compute the marginal probability density functions fU(u) and fV (v). (c) Are the random variables U and V orthogonal? (d) Are the random variables U and V independent?
