Spring98exam21

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Unformatted text preview: University of Illinois at Urbana-Champaign Department of Electrical and Computer Engineering Solutions to Hour Exam # 2 Problem 1 (18 points) True False p ECE 313 In the following statements, X and Y are generic continuous random variables with well-de ned nite mean and variance. The functions FX (u) and fX (u) denote the cumulative distribution function and the probability density function of X , respectively. p 2 p 2 2 2 2 p 2 2 2 2 p 2 p 2 2 fX (u) 1 for all u P fa < X < bg = P fea < eX < ebg for all real a and b P fX 2 ; 2X + 1 < ;1g = 0 If Z Poisson with parameter , then E (Z + 1)2] = ( + 1)2 If X and Y have the same variance, then E X 2] = E Y 2] If Y = jX j, then E Y ] E X ] Problem 2 (28 points) The received signal R in a communication system corresponds to one of two hypotheses: H0 : R = S0 H1 : R = S1 The random variables S0 and S1 are both Gaussian, with the same variance 2 = 4. It is known that E S0] = 0 while E S1] = 1. In other words, we have S0 N (0 22) and S1 N (1 22). The received signal R is now passed through a quantizer to produce one of the following three possible observations: A1 : R < ;1 A2 : ;1 R < +1 A3 : R +1 Some of the values of the cumulative distribution function ( ) of the standard Gaussian random variable N (0 1) are given below: x 0:0 0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 (x) 0:500 0:540 0:579 0:618 0:655 0:692 0:728 0:758 0:788 0:816 x 1:0 1:1 1:2 1:3 1:4 1:5 1:6 1:7 1:8 1:9 (x) 0:841 0:864 0:885 0:903 0:919 0:933 0:945 0:955 0:964 0:971 a. Calculate the probabilities of the events A1 A2 A3 under the two hypotheses, and complete the following likelihood matrix. Answer: The l...
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This note was uploaded on 06/15/2010 for the course ECE 313 taught by Professor Sarwate during the Spring '08 term at University of Illinois at Urbana–Champaign.

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