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Unformatted text preview: Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.432 Stochastic Processes, Detection and Estimation Problem Set 8 Spring 2004 Issued: Thursday, April 8, 2004 Due: Thursday, April 15, 2004 Reading: For this problem set: Chapter 5, Sections 6.1 and 6.3 Next: Chapter 6, Sections 7.1 and 7.2 Exam #2 Reminder: Our second exam will take place Thursday, April 22, 2004, 9am 11am. The exam will cover material through Lecture 16 (April 8) as well as the associated homework through Problem Set 8. You are allowed to bring two 8 1 × 11 sheets of notes (both sides). 2 Note that there will be no lecture on April 22. Problem 8.1 Consider the continuoustime process N x ( t ) = x i s i ( t ) , t T i =1 where the x i are zeromean and jointly Gaussian with E [ x i x j ] = µ ij , and the s i ( t ) are linearly independent, with T s n ( t ) s m ( t ) dt = α mn . (a) Calculate K xx ( t, ρ ). (b) Is K xx ( t, ρ ) positive definite? Justify your answer. (c) Find a matrix H whose eigenvalues are precisely the nonzero eigenvalues of the process x ( t ). Determine an expression for the elements of H in terms of µ ij and α mn . Hint: let N π ( t ) = b i s i ( t ) i =1 and introduce the vector notation b = [ b 1 b 2 ··· b N ] T ....
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 Spring '04
 Prof.GregoryWornell
 Electrical Engineering, Stochastic process, Massachusetts Institute of Technology, random process, Syy, Department of Electrical Engineering and Computer Science, Kyy

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