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# hw6 - UCSD ECE 153 Prof Young-Han Kim Handout#20 Thursday...

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UCSD ECE 153 Handout #20 Prof. Young-Han Kim Thursday, November 20, 2008 Homework Set #6 Due: Thursday, December 4, 2008 1. Read Sections 7.1–7.4, 9.1–9.6, 10.1–10.2, 10.4 in the text. Try to work on all examples. 2. Symmetric random walk. Let X n be a random walk defined as 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) Find P { max 1 i< 20 X i = 10 | X 20 = 0 } . (c) Find P { X n = k } . 3. Moving average process. Let Y n = 1 2 Z n - 1 + Z n for n 1 , where Z 0 , Z 1 , Z 2 , . . . are i.i.d. N (0 , 1). Find the mean and autocorrelation function of Y n . 4. Gauss-Markov process. Let X 0 = 0 and X n = 1 2 X n - 1 + Z n for n 1, where Z 1 , Z 2 , . . . are i.i.d. N (0 , 1). Find the mean and autocorrelation function of X n . 5. 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.

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hw6 - UCSD ECE 153 Prof Young-Han Kim Handout#20 Thursday...

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