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hw8 - EE 278 Statistical Signal Processing Homework#8 Due...

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EE 278 November 18, 2009 Statistical Signal Processing Handout #18 Homework #8 Due: Wednesday, December 2 1. Discrete-time Wiener process. Let { Z n : n 0 } be a discrete-time white Gaussian noise process; that is, Z 1 , Z 2 , Z 3 , . . . are i.i.d. N (0 , 1). Define the process { X n : n 0 } by 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 Gaussian process? Justify your answer. c. Find the mean and autocorrelation functions of X n . d. Specify the first-order pdf of X n . e. Specify the joint pdf of X 3 , X 5 , and X 8 . f. Find E( X 20 | X 1 , X 2 , . . . , X 10 ). g. Given X 1 = 4, X 2 = 2, and 0 X 3 4, find the minimum MSE estimate of X 4 . 2. Sawtooth process. Let X ( t ) = g ( t - T ), where g ( t ) is the periodic triangular waveform shown in Figure 1, and the delay T is a random variable with T U[0 , 1). . . . . . . . . 0 1 1 2 3 t g ( t ) Figure 1: Periodic triangular waveform Is X ( t ) a strict-sense stationary random process? Justify your answer. 3. Stationary Gauss-Markov process. (Bonus) Consider the following variation on the Gauss- Markov process discussed in the lecture notes 8: X 0 ∼ N (0 , a ) X n = 1 2 X n - 1 + Z n , n 1 , where Z 1 , Z 2 , Z 3 , . . . are i.i.d. N (0 , 1) independent of X 0 . a. Find a such that X n is stationary. Find the mean and autocorrelation functions of X n .

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hw8 - EE 278 Statistical Signal Processing Homework#8 Due...

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