4assignment

# 4assignment - Assignment 4 Problems 1 Let X denote a random variable distributed on the set A = cfw_a1 a2 aN with corresponding probabilities

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Assignment 4 Problems: 1. Let X denote a random variable distributed on the set A = { a 1 ,a 2 ,...,a N } with correspond- ing probabilities { p 1 ,p 2 ,...,p N } . Let Y be another random variable deﬁned on the same set but distributed uniformly. Show that H ( X ) H ( Y ) with equality if and only if X is also uniformly distributed. Hint: First prove the inequality log x x - 1 with equality for x = 1, then apply this inequality to N X n =1 p n log 1 N p n ! . 2. Determine the average energy of a set of M PAM signals of the form s m ( t ) = s m ψ ( t ) , m = 1 , 2 ,...,M where s m = p E g A m , m = 1 , 2 ,...,M The signals are equally probable with amplitudes that are symmetric about zero and are uniformly spaced with distance d between adjacent amplitudes as shownn in Figure 7.11. 3. Consider the four waveforms s 1 = (2 , - 1 , - 1 , - 1) s 3 = (1 , - 1 , 1 , - 1) s 2 = ( - 2 , 1 , 1 , 0) s 4 = (1 , - 2 , - 2 , 2) Determine the dimennsionality of the waveforms. Determine the minimum distance between any pair of vectors.

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## This note was uploaded on 11/13/2011 for the course ECEN 455 taught by Professor Staff during the Spring '08 term at Texas A&M.

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4assignment - Assignment 4 Problems 1 Let X denote a random variable distributed on the set A = cfw_a1 a2 aN with corresponding probabilities

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