ps8 - Massachusetts Institute of Technology Department of...

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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 continuous-time process N x ( t ) = x i s i ( t ) , t T i =1 where the x i are zero-mean 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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ps8 - Massachusetts Institute of Technology Department of...

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