it1_2012_u10m - Signal and Information Processing...

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Information Theory I Prof. Dr. A. Lapidoth Signal and Information Processing Laboratory Institut für Signal- und Informationsverarbeitung Model Answers to Exercise 10 of November 21, 2012 http://www.isi.ee.ethz.ch/teaching/courses/it1 Problem 1 Properties of R ( D ) We have the two rate distortion functions R ( D ) = min P ˆ X | X : E [ d ( x, ˆ x )] D I ( X ; ˆ X ) , and R ( D ) = min P ˆ X | X : E [ d ( x, ˆ x )] D I ( X ; ˆ X ) , where the distortion measures d ( · , · ) and d ( · , · ) relate through d ( i,j ) = d ( ) w i . We will show that R ( D ) = R ( D + ¯ w ) by expressing E b d ( X, ˆ X ) B in terms of E b d ( X, ˆ X ) B : E b d ( X, ˆ X ) B = m s i =1 m s j =1 p ( ) ( d ( ) w i ) = m s i =1 m s j =1 p ( ) d ( ) m s i =1 m s j =1 p ( ) w i = E b d ( X, ˆ X ) B m s i =1 p i w i m s j =1 p ( j | i ) ± ²³ ´ =1 = E b d ( X, ˆ X ) B ¯ w. Hence, we have R ( D ) = min P ˆ X | X : E [ d ( X, ˆ X ) ] D I ( X ; ˆ X ) = min P ˆ X | X : E [ d ( X, ˆ X ) ] ¯ w D I ( X ; ˆ X ) = min P ˆ X | X : E [ d ( X, ˆ X ) ] D + ¯ w I ( X ; ˆ X ) = R ( D + ¯ w ) . To conclude that there is no loss in generality in assuming that min j d ( ) = 0 for all i { 1 , 2 ,... ,m } , it remains to note that for every distortion measure d ( · , · ), there exists a corre- sponding distortion measure d ( · , · ) with min j d ( ) = 0 for every i ∈ { 1 , 2 } (by choosing w i = min j d ( )). And as we have shown, the rate distortion function R ( · ) then follows directly from R ( · ) by R ( D ) = R ( D ¯ w ) . c c Amos Lapidoth, 2012 1
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Problem 2 Erasure Distortion Note frst that For rate R = 0 the distortion D = 1 is achievable through the choice P ˆ X | X (“?” | x ) = 1. ±urthermore, d (1 , 0) = and d (0 , 1) = imply that the rate distortion Function is certainly only achieved by conditional laws satisFying Pr b ˆ X = 0 v v v X = 1 B = 0 and Pr b ˆ X = 1 v v v X = 0 B = 0, since otherwise the expected distortion would be infnite. It Follows that
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