PS0 10

# PS0 10 - University of Minnesota Dept of Electrical and...

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University of Minnesota Dept. of Electrical and Computer Engineering EE 8581 DETECTION AND ESTIMATION THEORY Spring 2010 Problem Set 0 Assigned: January 26, 2010 Due: February 2, 2010 Problem 1 This problem is designed to develop the geometric approach to detection and estimation problems. Given three zero-mean random variables x 1 , x 2 and x 3 with covariance matrix: = Λ 1 1 1 23 13 23 12 13 12 λ λ λ λ λ λ . a. Find a random variable z 2 = x 2 + a 1 x 1 such that z 2 is orthogonal to x 1 . b. Find another variable z 3 = x 3 + b 2 x 2 + b 1 x 1 that is orthogonal to x 1 and x 2 . This process is equivalent of the Gram-Schmidt procedure is linear algebra. c. Any random variable y is said to lie in the subspace of x 1 and x 2 if y = c 2 x 2 + c 1 x 1 . Under what condition is x 3 in the subspace of x 1 and x 2 ? Formulate your answer in terms of the elements of the covariance matrix Λ . (Remember that in our view, a random variable x has zero length if E(x 2 ) = 0.) Problem 2

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