mid1sol - 1 National Chiao Tung University Department of...

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Unformatted text preview: 1 National Chiao Tung University Department of Electronics Engineering Stochastic Processes, Midterm I, 2006 Solutions NOTE: • Remember to write down your name. • There are 5 problem sets with 100 points in total. 1. True or False (5+5+5+5=20 points) Please provide an explanation, either a simple proof or a counter example, to your answer. Otherwise, your answer will carry no credits. (a) The eigenvalue of a covariance matrix is always non-negative. True. For any covariance matrix K , we can rewrite it as K = E [ xx H ] with appropriate x . Then, for the eigenvector e of K associated with eigenvalue λ , we have E [ xx H ] e = λ e . Let-multiplying both sides by e H gives e H E [ xx H ] e = λ e H e = λ || e || 2 . Since the left-hand side of the above is non-negative, the right-hand side is also the case, implying λ ≥ . (b) Let X and Y be two arbitrary zero mean random variables. Then, X and Y- E [ Y | X ] are uncorrelated. True. To justify this, we need to show E [ X ( Y- E [ Y | X ])] = E [ X ] E [ Y- E [ Y | X ]] . The correlation is E [ X ( Y- E [ Y | X ])] = E [ XY- XE [ Y | X ]] = E [ XY- E [ XY | X ]] = E [ XY ]- E [ E [ XY | X ]] = E [ XY ]- E [ XY ] = 0 , which equals to E [ X ] E [ Y- E [ Y | X ]] (since X is zero mean). (c) Suppose E [ XY ] = E [ X ] E [ Y ] for two Gaussian random variables X and Y . Then, X and Y are independent. False. X and Y have to be jointly Gaussian. (d) An eigenvalue of the matrix AA H is also an eigenvalue of A H A for any nonzero matrix A . True. See HW1 Prob. 4-(a). 2 2. Singular Value Decomposition (10 points) Find the singular value decomposition of the matrix A = • 1 √ 3 2 ‚ ....
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This note was uploaded on 11/28/2010 for the course EE 301 taught by Professor Gfung during the Winter '10 term at National Chiao Tung University.

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mid1sol - 1 National Chiao Tung University Department of...

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