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UCSD ECE287B
Handout #3
Prof. YoungHan 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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