Hw3Soln - ECE 5670 Digital Communications ECE department...

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Unformatted text preview: ECE 5670 : Digital Communications ECE department, Cornell University, Spring 2011 Homework 3 Solutions Instructor: Salman Avestimehr Office 325 Rhodes Hall 1. (a) Assume the first codeword is transmitted. P [ success ] = P ( y [1] > y [ j ] ,j = 2 , , 2 B ) total probability law = integraldisplay f y [1] ( a ) P ( y [1] > y [ j ] ,j = 2 , , 2 B | y [1] = a ) da = integraldisplay f w [1] ( a- radicalbig BE b ) P ( y [1] > y [ j ] ,j = 2 , , 2 B | y [1] = a ) da = integraldisplay f w [1] ( a- radicalbig BE b ) 2 B productdisplay j =2 P ( a > w [ j ]) da = integraldisplay f ( a- BE b )(1- Q ( a )) 2 B 1 da a = a = integraldisplay f ( a- BE b )(1- Q ( a )) 2 B 1 d a Therefore, P [ E ] = 1- integraldisplay f ( a- BE b )(1- Q ( a )) 2 B 1 da = integraldisplay f ( a- BE b ) da- integraldisplay f ( a- BE b )(1- Q ( a )) 2 B 1 da = integraldisplay f ( a- BE b ) parenleftBig 1- (1- Q ( a )) 2 B 1 parenrightBig da (b) Since 1- Q ( a ) , 1- (1- Q ( a )) 2 B 1 1 . Also since (1- x ) m 1- mx for x 1 we have 1- (1- Q ( a )) 2 B 1 1- (1- (2 B- 1) Q ( a )) = (2 B- 1) Q ( a ) (2 B- 1) e a 2 2 2 B e a 2 2 (c) P [ E ] = integraldisplay f parenleftbigg a- BE b parenrightbigg parenleftBig 1- (1- Q ( a )) 2 B 1 parenrightBig da. = integraldisplay a th f parenleftbigg a- BE b parenrightbigg parenleftBig 1- (1- Q ( a )) 2 B 1 parenrightBig da + integraldisplay a th f parenleftbigg a- BE b parenrightbigg parenleftBig 1- (1- Q ( a )) 2 B...
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This note was uploaded on 10/02/2011 for the course ECE 5670 taught by Professor Scaglione during the Spring '11 term at Cornell.

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Hw3Soln - ECE 5670 Digital Communications ECE department...

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