7.26 - S-72.2410 Information Theory Haanpaa &...

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Unformatted text preview: S-72.2410 Information Theory Haanpaa & Linja-aho Homework 4, solutions 2009 Homework 4, solutions Deadline: November 30th, 16:00 The box for returning exercises is in the E-wing, 2nd floor corridor. 1. Consider a channel a45 a45 a63 + X Y Z X = { , 1 , 2 , 3 } , where Y = X + Z , and Z is uniformly distributed over three distinct integer values Z = { z 1 ,z 2 ,z 3 } . (a) What is the maximum capacity over all choices of the Z alphabet? Give distinct integer values z 1 ,z 2 ,z 3 and a distribution on X achieving this. (b) What is the minimum capacity over all choices for the Z alphabet? Give distinct integer values z 1 ,z 2 ,z 3 and a distribution on X achieving this. Solution: (a) The maximum capacity can be achieved by choosing the noise alphabet so that the values are so far away from each other that they do not disturb the transmission. For example, if we choose Z = { 100 , 200 , 300 } , no two different input alphabet result in same output alphabet. The capacity of the channel is now C = 2 bits and it can be achieved by using a uniform distribution at the input p ( X ) = { 1 4 , 1 4 , 1 4 , 1 4 } . (b) In a-part we tried to minimize the overlapping of output symbols, now we are trying to maximize them. Maximal overlapping can be achieved with values Z = { , 1 , 2 } . Now, the capacity C = 1 bit. It can be achieved with input distribution p ( X ) = { 1 2 , , , 1 2 } . 2. Consider the channel with x,y , 1 , 2 , 3 and transition probabilities p ( y | x ) given by the following matrix: 1 2 1 2 0 0 1 2 1 2 0 0 1 2 1 2 1 2 0 0 1 2 Page 1 of 5 S-72.2410 Information Theory Haanpaa & Linja-aho Homework 4, solutions 2009 (a) Find the capacity of this channel.Find the capacity of this channel....
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7.26 - S-72.2410 Information Theory Haanpaa &...

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