hw7 - UCSD ECE153 Prof Young-Han Kim Homework Set#7 Due...

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UCSD ECE153 Handout #33 Prof. Young-Han 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 = 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. Discrete-time 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 = 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 0 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 0 . (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 ). (e) Find the MMSE estimate of X 100 given ( X 101 , X 102 , . . . , X 199 ). (f) Find the MMSE estimate of X 100 given ( X 1 , . . . , X 99 , X 101 , . . . , X 199 ).
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