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# hw8 - Virginia Tech The Bradley Department of Electrical...

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Virginia Tech The Bradley Department of Electrical and Computer Engineering ECE 5605 – Stochastic Signals and Systems – Fall 08 Problem Set 8 Problem 1. Let U n be a sequence of i.i.d. zero-mean, unit-variance Gaussian random variables. A “low-pass filter” takes the sequence U n and produces the sequence X n = 1 2 U n + 1 2 2 U n - 1 + . . . + 1 2 n U 1 (a) Find the mean and the variance of X n . (b) Find the characteristic function of X n . What happens as n approaches infinity? Problem 2. Let Y n be the process that results when individual 1s in a Bernoulli process are erased with probability α . Find the pmf of S n , the counting process for Y n . Does Y n have independent and stationary increments? Problem 3. Find P [ S n = 0] for the random walk process. Assume P [ X n = 1] = P [ X n = - 1] = 1 / 2. Problem 4. Consider the following moving average process Z n = 2 3 X n + 1 3 X n - 1 X 0 = 0 Find the mean, variance, and covariance of Z n if X n is a Bernoulli random process.

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