spring98exam2[1]

Spring98exam21

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online