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Unformatted text preview: UCSD ECE153 Handout #33 Prof. YoungHan Kim Thursday, May 26, 2011 Homework Set #7 Due: Thursday, June 2, 2011 1. Symmetric random walk. Let X n be a random walk defined by X = 0 , X n = n summationdisplay i =1 Z i , where Z 1 ,Z 2 ,... are i.i.d. with P { Z 1 = 1 } = P { Z 1 = 1 } = 1 2 . (a) Find P { X 10 = 10 } . (b) Approximate P { 10 ≤ X 100 ≤ 10 } using the central limit theorem. (c) Find P { X n = k } . 2. Discretetime Wiener process. Let Z n , n ≥ 0 be a discrete time white Gaussian noise (WGN) process, i.e., Z 1 ,Z 2 ,... are i.i.d. ∼ N (0 , 1). Define the process X n , n ≥ 1 as X = 0, and X n = X n 1 + Z n for n ≥ 1. (a) Is X n an independent increment process? Justify your answer. (b) Is X n a Markov process? Justify your answer. (c) Is X n a Gaussian process? Justify your answer. (d) Find the mean and autocorrelation functions of X n . (e) Specify the first and second order pdfs of X n . (f) Specify the joint pdf of X 1 ,X 2 , and X 3 . (g) Find E ( X 20  X 1 ,X 2 ,...,X 10 ). 3. Arrow of time. Let X be a Gaussian random variable with zero mean and unit variance, and X n = αX n 1 + Z n for n ≥ 1, where α is a fixed constant with  α  < 1 and Z 1 ,Z 2 ,... are i.i.d. ∼ N (0 , 1 α 2 ), independent of X . (a) Is the process { X n } Gaussian? (b) Is { X n } Markov? (c) Find R X ( n,m ). (d) Find the (nonlinear) MMSE estimate of X 100 given ( X 1 ,X 2 ,...,X 99 )....
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 Stochastic process, yn, Stationary process, Xn

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