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hw6 - EE 378 Statistical Signal Processing Homework Set#6...

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EE 378 Handout #13 Statistical Signal Processing Wednesday, May 16, 2007 Homework Set #6 Due: Wednesday, May 23, 2007. Announcement: You can hand in the HW either after class or deposit it, before 5pm, in the Homework in box in the 378 drawer of the class file cabinet on the second floor of the Packard Building. 1. The following graph (Fig. 1) represents the dependencies between 6 random variables. As discussed in class, an edge between random variables means that the joint pmf can be written as a factor of products involving pairs X i and X j for the random variables X i and X j that are connected in the graph. Figure 1: Problem 1 X1 X2 X3 X4 X5 X6 Using the graph, state whether or not the statements given below “necessarily” hold true. (a) X 1 and X 6 are conditionally independent given X 2 and X 5 . (b) X 1 and X 6 are conditionally independent given X 3 and X 4 . (c) X 2 and X 3 are independent. (d) X 2 and X 3 are conditionally independent given X 4 . 1
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2. Let X ( n + 1) = aX ( n ) + bW ( n ) Z ( n ) = cX ( n ) Y ( n ) = Z ( n ) + V ( n ) , where { V ( n ) } and { W ( n ) } are zero-mean, unit-variance white noise processes, inde- pendent of each other, and a = 0 . 5, b = 0 . 2, and c = 0 . 2. (a) Draw a block diagram of the model. (b) Find the causal Wiener filter for estimating Z ( n ) based on Y ( k ), -∞ < k n . (c) Find the asymptotic value of K ( n ) of the Kalman filter as n approaches .
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