oldfinal2sol

# oldfinal2sol - UCSD ECE153 Prof Young-Han Kim Handout#37...

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3. Stationary process (20 points). Consider the Gaussian autoregressive random process X k +1 = 1 3 X k + Z k , k = 0 , 1 , 2 ,..., where Z 0 ,Z 1 ,Z 2 ,... are i.i.d. N (0 , 1). (a) Find the distribution on X 0 that makes this a stationary stochastic process. (b) What is the resulting autocorrelation R X ( n )? Solution: (a) Consider X 0 N (0 , 9 / 8). It is easy to see that X n N (0 , 9 / 8), which implies that X n is stationary (why?). (b) We have X n = 1 3 X n - 1 + Z n - 1 = p 1 3 P 2 X n - 2 + 1 3 Z n - 2 + Z n - 1 = p 1 3 P n X 0 + n s k =1 p 1 3 P n - k Z k - 1 . Hence, R X ( n ) = EX 0 X n = p 9 8 Pp 1 3 P n for n 0, and in general R X ( n ) = EX 0 X n = p 9 8 Pp 1 3 P | n | .
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oldfinal2sol - UCSD ECE153 Prof Young-Han Kim Handout#37...

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