midterm2008_solution

midterm2008_solution - School of Computer and Communication...

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´ ´ ´ School of Computer and Communication Sciences Principles of Digital Communications: Assignment date: April 17, 2008 Summer Semester 2009 Due date: April 17, 2008 Solution Midterm Exam Problem 1. Playing Roulette Let N b and N r be the number of black and red numbers, respectively, in a sequence y = y 1 y 2 . . . y K . Let H = 0 denote a hypothesis that a table is normal, and H = 1 that it is loaded. (a) On a normal table, each number can be an outcome with probability 1 37 . Hence, p Y | H ( y | 0) = parenleftbigg 1 37 parenrightbigg K On a loaded table, each black number appears with probability 3 2 1 37 , and each red number appears with probability 1 2 1 37 . Therefore, p Y | H ( y | 1) = parenleftbigg 1 37 parenrightbigg K parenleftbigg 3 2 parenrightbigg N b parenleftbigg 1 2 parenrightbigg N r . (b) MAP rule minimizes the probability of error: p H (0) p Y | H ( y | 0) ˆ H =0 ˆ H =1 p H (1) p Y | H ( y | 1) , which can be reduced to ln 9 ˆ H =0 ˆ H =2 N b ln 3 2 + N r ln 1 2 . Problem 2. (Binary Communication Across the Vector Gaussian Channel with Non-Identity Covariance Matrix) (a) As we have seen in class, the optimal decision regions for the vector Gaussian channel with identity covariance matrix of the noise are the Veronoi regions of the transmission points.
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a b a b X (0) y 1 y 2 R 0 R 1 X (1) y = » b a b a (b) Yes, because the map from ( y 1 , y 2 ) to ( w 1 , w 2 ) is reversible. Given ( w 1 , w 2 ), we know that MAP decoding is the optimal strategy. Given ( w 1 , w 2 ), another option is to recover the original ( y 1 , y 2 ) and then apply a MAP decision rule. MAP decoding from ( w 1 , w 2 ) should perform at least as good as the second method which we know is the optimal strategy to decode H from ( y 1 , y 2 ).
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