hw2 soln

# hw2 soln - Dr Behnaam Aazhang ELEC 430 Department of...

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Dr. Behnaam Aazhang ELEC 430 Department of Electrical and Computer Engineering Rice University Due 24 Jan 2008 HOMEWORK 2 — Random Processes Exercise 1. Linear Transformations of Random Processes Suppose A and B are two Gaussian random variables, each zero mean with E ± A 2 ² < and E ± B 2 ² < . The correlation between them is denoted by E [ AB ]. Deﬁne the random processes X t = A + Bt and Y t = B + At . (a) Find the mean, autocorrelation, and crosscorrelation of X t and Y t . (b) Find the 1st order density of X t , f X t ( x ). (c) Find the conditional density of X t 2 given X t 1 , denoted f X t 2 | X t 1 ( x 2 | x 1 ). Assume that t 2 > t 1 . Hint: See Proakis and Salehi, problem 3.28. (d) Is X t wide sense stationary? Exercise 2. Stationarity Show that if X t is second-order stationary, then it is also ﬁrst-order stationary. Exercise 3. Probability Distributions Let the stochastic process X t be deﬁned by X t = cos( Wt +Θ) where W and Θ are statistically independent random variables. Θ is uniformly distributed over [

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## This homework help was uploaded on 02/11/2008 for the course ELEC 430 taught by Professor Aazhang during the Spring '08 term at Rice.

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hw2 soln - Dr Behnaam Aazhang ELEC 430 Department of...

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