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m1_sol - ECE 562 Fall 2011 EXAM 1 Solutions 1(15 pts total...

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ECE 562 Fall 2011 October 12, 2011 EXAM 1 Solutions 1. (15 pts total) True or False. Determine if the following statements are True or False. You need to provide a brief justification for your answer to get credit. (a) The complex baseband representation of 2 cos( πt ) sin(2 πf c t ) is cos( πt ). Ans: False. 2 cos( πt ) sin(2 πf c t ) = 2 cos( πt ) cos(2 πf c t - π/ 2). Thus, by the defini- tion, the complex baseband representation is cos( πt ) e - jπ/ 2 = - j cos( πt ). (b) If X and Y are zero-mean proper complex random variables, then Z = X + Y is also proper complex. Ans: False. Clearly E [ Z ] = 0. Therefore the pseudo-variance of Z is given by E [ Z 2 ], which equals E [ X 2 ]+ E [ Y 2 ]+2 E [ XY ]. By properness of X and Y and the fact that they are zero mean, we have that E [ X 2 ] = E [ Y 2 ] = 0. Thus E [ Z 2 ] = 2 E [ XY ]. Thus E [ Z 2 ] = 0, unless [ X Y ] is a proper complex random vector or if X and Y are independent. As a counterexample, consider X , Y which are CN (0 , 1) random variables, with E [ XY ] = 2. (c) In a linear modulation constellation, if three nearest neighbors form an equilateral triangle, it is impossible to Gray code the constellation. Ans: True. If we make both neighbors of a point differ by one bit from it, the neighbors have to differ by two bits from each other.
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