sol3 - Boston University Department of Electrical and...

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Boston University Department of Electrical and Computer Engineering ENG EC515 Digital Communication ( Fall 2009 ) Solution to Problem set 3 Posted: Mon 5 Oct Suggested reading: Class-notes, handouts, and Ch. 4 textbook Problem 3.1 (Optimum receiver) [4pts] A communication system is used to transmit one of two equally likely messages, M ∈ M = { 1 , 2 } . The channel output is a voltage (a continuous random variable), Y R , whose conditional density function is given by: p Y | M ( y | 1) = y 0 y 1 2 - y 1 y 2 0 otherwise p Y | M ( y | 2) = ( 0 . 25 - 1 y 3 0 otherwise (a) [2pts] Determine b m opt ( y ) the optimum (minimum probability of error) receiver decision rule. (b) [2pts] Compute P e the resulting probability of error. Solution: [4pts] We have the following p M (1) = p M (2) = 0 . 5. Hence, b m opt ( y ) = b m MAP ( y ) = b m ML ( y ) = arg max m ∈{ 1 , 2 } p Y | M ( y | m ) = 1 if p Y | M ( y | 1) p Y | M ( y | 2) 2 otherwise. From the figure, p Y | M ( y | 2) y 0 - 1 1 3 2 1 p Y | M ( y | 1) (1 . 75 , 0 . 25) (0 . 25 , 0 . 25) (a) [2pts] b m opt ( y ) = 1 if 0 . 25 y 1 . 75 2 otherwise . (b) [2pts] P e = 1 2 × Shaded area = 1 2 × 1 2 × 0 . 25 × (2 + 1 . 5) = 1 16 × 7 2 = 7 32 1
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Problem 3.2 ( Value of noisy look at noise) [12pts] A communication system is used to transmit one of two equally likely messages, M ∈ M = { 1 , 2 } . The channel inputs are voltage levels (real numbers). The channel encoder uses a real channel codebook consisting of two voltage levels: C = { x 1 , x 2 R : x 2 = - x 1 = E s } . At the output of the channel we receive a pair of (random) voltage levels given by: Y 1 = X + Z 1 Y 2 = Z 1 + Z 2 where, X = x M is the transmitted voltage level and Z 1 , Z 2 are noise voltages that are independent of each other and the message and are Gaussian distributed with zero mean and variance σ 2 .
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