PS6 10 - where the x i are zero-mean and jointly Gaussian...

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University of Minnesota Dept. of Electrical and Computer Engineering EE 8581 DETECTION AND ESTIMATION THEORY Spring 2010 Problem Set 6 Assigned: March 23, 2010 Due: March 30, 2010 Readings : Read Levy Chapters 3 and 6. Problems : Solve problems 6.2, 6.4 and 6.7 in Chapter 6 of Levy’s book and the following problems: Problem 4: 1. Prove that λ =0 and λ = 2P / α cannot be eigenvalues of the covariance function Pe - α |t-u| over the interval [-T,T]. 2. Show that the largest λ 1 eigenvalue of K(t,u) over the interval [-T,T] satisfies the inequality ∫ ∫ T T T T dtdu u f u t K t f ) ( ) , ( ) ( 1 λ for any function f(t) with unit energy in the interval [-T,T]. Problem 5: Consider the process x(t) = i=1 N x i S i (t) 0 ≤ t ≤T
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Unformatted text preview: where the x i are zero-mean and jointly Gaussian with E[x i x j ]= µ ij and ⌡ ⌠ T S n (t) S m (t) dt = ρ nm . Assume that the S i (t) are linearly independent. a) Compute K x (t,u). b) Is K x (t,u) positive definite? Explain your answer. c) Find a matrix H whose eigenvalues are precisely the nonzero eigenvalues of K x (t,u). Determine an expression for the elements of H in terms of µ ij and ρ nm . Hint: let λ ≠0 and let φ (t) = ∑ i=1 n b i S i (t) Introduce the vector notation b = b 1 b 2 . . . b N Show that if φ (t) is an eigenfunction of K x (.,.) with eigenvalue λ , then λ b = H b...
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This note was uploaded on 06/04/2010 for the course EE 8581 taught by Professor Staff during the Spring '08 term at Minnesota.

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PS6 10 - where the x i are zero-mean and jointly Gaussian...

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