hw1278b - UCSD ECE287B Prof. Young-Han Kim Homework Set #1...

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UCSD ECE287B Handout #3 Prof. Young-Han Kim Thursday, September 30, 2010 Homework Set #1 Due: Thursday, October 7, 2010 1. Provide the details of the proof of Theorem 15.2. 2. Common information. Consider the optimal rate region R * of the Gray–Wyner system in Theo- rem 15.3. (a) Complete the derivations of the three measures of common information as extreme points of R * . (b) Show that the three measures of common information satisfy 0 K ( U 1 ; U 2 ) I ( U 1 ; U 2 ) J ( U 1 ; U 2 ) H ( U 1 ,U 2 ) . (c) Show that K ( U 1 ; U 2 ) = I ( U 1 ; U 2 ) = J ( U 1 ; U 2 ) iF U 1 = ( V,V 1 ) and U 2 = ( V,V 2 ) for some V 1 V V 2 . 3. Lossy Gray–Wyner system. Consider the Gray–Wyner system in Section 15.2 with two distortion measures d 1 ,d 2 . The sources U 1 and U 2 are to be recovered with distortions D 1 and D 2 , respectively. Show that the rate–distortion region R ( D 1 ,D 2 ) is the set of rate pairs ( R 1 ,R 2 ) such that R 0 I ( U 1 ,U 2 ; V ) , R 1 I
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