S-72_2410_solution_5

S-72_2410_solution_5 - S-72.2410 Information Theory...

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Unformatted text preview: S-72.2410 Information Theory Haanp¨a¨a & Linja-aho Homework 5, solutions 2009 Homework 5, solutions Deadline: December 7th, 16:00 The box for returning exercises is in the E-wing, 2nd floor corridor. 1. Let Y 1 and Y 2 be conditionally independent and conditionally identi- cally distributed given X . (a) Show I ( X ; Y 1 ,Y 2 ) = 2 I ( X ; Y 1 )- I ( Y 1 ; Y 2 ). (b) Conclude that the capacity of the channel a45 a45 X ( Y 1 ,Y 2 ) is less than twice the capacity of the channel a45 a45 X Y 1 Solution: a) I ( X ; Y 1 ,Y 2 ) = H ( Y 1 ,Y 2 )- H ( Y 1 ,Y 2 | X ) and since Y 1 and Y 2 are conditionally independent for given X = H ( Y 1 ) + H ( Y 2 )- I ( Y 1 ; Y 2 )- H ( Y 1 | X )- H ( Y 2 | X ) = I ( X ; Y 1 ) + I ( X ; Y 2 )- I ( Y 1 ; Y 2 ) and since Y 1 and Y 2 are conditionally identically distributed: = 2 I ( X ; Y 1 )- I ( Y 1 ; Y 2 ) b) The capacity of the single look channel X → Y 1 is C 1 = max p ( x ) I ( X ; Y 1 ) . Page 1 of 4 S-72.2410 Information Theory Haanp¨a¨a & Linja-aho Homework 5, solutions 2009 The capacity of the channel X → ( Y 1 ,Y 2 ) is C 2 = max p ( x ) I ( X ; Y 1 ,Y 2 ) = max p ( x ) 2 I ( X ; Y 1 )- I ( Y 1 ; Y 2 ) ≤ max p ( x ) 2 I ( X ; Y 1 ) = 2 C 1 ....
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This note was uploaded on 10/27/2010 for the course ECE 221 taught by Professor Sd during the Spring '10 term at Huston-Tillotson.

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S-72_2410_solution_5 - S-72.2410 Information Theory...

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