Hw1Soln - ECE 5670 Digital Communications ECE department...

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ECE 5670 : Digital Communications ECE department, Cornell University, Spring 2011 Homework 1 Solutions Instructor: Salman Avestimehr Office 325 Rhodes Hall 1. (a) E [ N ]= X n =0 n Pr( N = n ) = X n =0 n (Pr( N>n 1) Pr( )) = X n =0 n Pr( 1) X n =0 n Pr( ) = X n =0 ( n +1)Pr( ) X n =0 n Pr( ) = X n =0 Pr( ) (b) Z 0 Pr( X a ) da = Z 0 Z a f X ( x ) dx da = Z 0 Z x 0 f X ( x ) da dx = Z 0 xf X ( x ) dx = E [ X ] (c) E [ X Z 0 xf X ( x ) dx = Z a 0 xf X ( x ) dx + Z a xf X ( x ) dx Z a xf X ( x ) dx Z a af X ( x ) dx = a Pr( X a )
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Decide 2 12 3 4 0 1 2 3 4 Decide +2 Figure 1: Distinct voltage points and the decision regions without the knowledge of the shape of noise pdf. (d) De±ne Y =( X E [ X ]) 2 By using (c) we have Pr( | X E [ X ] |≥ b )=P r ( Y b 2 ) E [ Y ] b 2 = E [ | X E [ X ] | 2 ] b 2 = σ 2 X b 2 2. (a) In the terminology of the class notes E =2and σ = 2. Hence the number of distinct voltage points that can be reliably transmitted is b E σ c +1=2. Figure1 shows the two voltage points along with the decision region. (b) This example shows that the knowledge of the pdf can be useful in packing more voltage points with the same energy constraint. We can exploit the speci±c shape
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Hw1Soln - ECE 5670 Digital Communications ECE department...

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