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midterm2007 - Principles of Digital Communications EPFL...

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Principles of Digital Communications EPFL Summer Semester 2007 April 26, 2007 Midterm This is a closed-book exam, but a single page of hand-written notes is allowed. Cal- culators and cell-phones are not allowed. You have 1 hour 45 minutes to solve the problems. Try to do side calculations on a separate sheet and report well organized solutions. If we can’t read it we can’t grade it. If you need additional pages, do not forget to write your name! Good Luck!! Name: Problem 1 / 13 Problem 2 / 5 Problem 3 / 6 Problem 4 / 11 Total / 35 1
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1. Bhattacharyya Bound (13 points) When Y R is an continuous random variable, the Bhattacharyya bound states that Pr { Y ∈ B i,j | H = i } ≤ radicalBigg P H ( j ) P H ( i ) integraldisplay y R radicalBig f Y | H ( y | i ) f Y | H ( y | j ) dy, where i, j are two possible hypotheses and B i,j = { y R : P H ( i ) f Y | H ( y | i ) P H ( j ) f Y | H ( y | j ) } . In this problem H = { 0 , 1 } and P H (0) = P H (1) = 0 . 5. (a) (2 points) Write a sentence that expresses the meaning of Pr { Y ∈ B 0 , 1 | H = 0 } . (You may write in English or French. Use words that have operational meaning.)
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