CHONG AND LIANG: CAPACITY REGION OF THE PARALLEL DEGRADED BROADCAST CHANNEL 5021
I
T
1
; Z
N
|T
0
+ I
X
N
;Y
N
|T
0
, T
1
≤
N
n=1
I
T
1
,
˜
Z
n−1
1
; Z
n
|T
0
, Z
N
n+1
+
N
n=1
I
X
n
;Y
n
|T
0
, T
1
, Z
N
n+1
,
˜
Z
n−1
1
Proof: Refer to Appendix A.
Remark 2: We note that the following Markov chains,
X → Y → Z and X → Y →
˜
Z, must hold in order to
prove Lemma 1. The Csiszár-sum identity alone is insufficient
to prove Lemma 1 for all distributions on
X, Y, Z,
˜
Z
.
We can now state the capacity theorem for the first class
of three-user parallel degraded BCs:
Theorem 2: The capacity region, C
π,1
, of a discrete mem-
oryless three-receiver BC with three-degraded message sets,
consisting of the product of five parallel degraded BCs given
by (1)-(5), is given by R
π
in Corollary 1 with X
6
∅.
Proof: The bulk of the proof (and the more interesting
part) lies in the converse. The identification of the auxiliary
random variables is quite standard in the proof of converses for
product channels. We will focus on the proof for (23) and (25).
The proof for the rest of the inequalities follow accordingly
and is given in Appendix B.
To prove the inequality (23), we consider the following
inequalities:
I
W
1
; M
N
1
+ I
W
2
, W
3
; Z
N
1
|W
1
≤
N
n=1
I
W
1
, M
n−1
1,1
; M
1,n
+
N
n=1
I
X
1,n
; Z
1,n
|W
1
, M
n−1
1,1
. (31)
I
W
1
; M
N
2
|M
N
1
+ I
W
2
, W
3
; Z
N
2
|W
1
, Z
N
1
≤
N
n=1
I
W
1
, M
N
1
, M
n−1
2,1
; M
2,n
+
N
n=1
I
X
2,n
; Z
2,n
|W
1
, M
N
1
, M
n−1
2,1
. (32)
I
W
1
; M
N
3
|M
N
(1→2)
+ I
W
2
, W
3
; Z
N
3
|W
1
, Z
N
(1→2)
≤ I
W
1
, M
N
(1→2)
; M
N
3
+ I
X
N
3
; M
N
3
|W
1
, M
N
(1→2)
−I
W
2
, W
3
; M
N
3
|W
1
, Z
N
(1→3)
= I
X
N
3
; M
N
3
− I
W
2
, W
3
; M
N
3
|W
1
, Z
N
(1→3)
≤
N
n=1
I
X
3,n
; M
3,n
− I
W
2
, W
3
; M
N
3
|W
1
, Z
N
(1→3)
(33)
I
W
1
; M
N
4
|M
N
(1→3)
+ I
W
2
, W
3
; Z
N
4
|W
1
, Z
N
(1→3)
≤ I
W
1
, M
N
(1→3)
; M
N
4
+ I
X
N
4
; M
N
4
|W
1
, M
N
(1→3)
+ I
W
2
, W
3
; M
N
3
|W
1
, Z
N
(1→3)
− I
W
2
, W
3
; M
N
(3→4)
|W
1
, Z
N
(1→4)
= I
X
N
4
; M
N
4
+ I
W
2
, W
3
; M
N
3
|W
1
, Z
N
(1→3)
− I
W
2
, W
3
; M
N
(3→4)
|W
1
, Z
N
(1→4)
≤
N
n=1
I
X
4,n
; M
4,n
+ I
W
2
, W
3
; M
N
3
|W
1
, Z
N
(1→3)
− I
W
2
, W
3
; M
N
(3→4)
|W
1
, Z
N
(1→4)
(34)
I
W
1
; M
N
5
|M
N
(1→4)
+ I
W
2
, W
3
; Z
N
5
|W
1
, Z
N
(1→4)
≤ I
W
1
, Y
N
1
, Z
N
2
, M
N
(3→4)
; M
N
5
+ I
X
N
5
; Z
N
5
|W
1
, Y
N
1
, Z
N
2
, M
N
(3→4)
+ I
W
2
, W
3
; M
N
(3→4)
|W
1
, Z
N
(1→4)
(a)
≤
N
n=1
I
W
1
, Y
N
1
, Z
N
2
, M
N
(3→4)
, M
n−1
5,1
, Z
N
5,n+1
; M
5,n
+
N
n=1
I
X
5,n
; Z
5,n
|W
1
, Y
N
1
, Z
N
2
, M
N
(3→4)
, M
n−1
5,1
, Z
N
5,n+1
+ I
W
2
, W
3
; M
N
(3→4)
|W
1
, Z
N
(1→4)
. (35)
where (a) follows from the Csiszár-sum identity.
Combining (31) to (35), we obtain
I
W
1
; M
N
(1→5)
+ I
W
2
, W
3
; Z
N
(1→5)
|W
1
≤
N
n=1
I
U
1,n
; M
1,n
+
N
n=1
I
X
1,n
; Z
1,n
|U
1,n
+
N
n=1
I
U
2,n
; M
2,n
+
N
n=1
I
X
2,n
; Z
2,n
|U
2,n
+
N
n=1
I
X
3,n
; M
3,n
+
N
n=1
I
X
4,n
; M
4,n
+
N
n=1
I
U
5,n
; M
5,n
+
N
n=1
I
X
5,n
; Z
5,n
|U
5,n
where
U
1,n
W
1
, M
n−1
1,1
(36)
U
2,n
W
1
, M
N
1
, M
n−1
2,1
(37)
U
5,n
W
1
, Y
N
1
, Z
N
2
, M
N
(3→4)
, M
n−1
5,1
, Z
N
5,n+1
. (38)
To prove the inequality (25), we consider the following
inequalities:
I
W
1
; M
N
1
+ I
W
2
;Y
N
1
|W
1
+ I
W
3
; Z
N
1
|W
1
, W
2
≤
N
n=1
I
W
1
, M
n−1
1,1
; M
1,n
+
N
n=1
I
W
1
, W
2
, Y
n−1
1,1
, M
n−1
1,1
;Y
1,n
|W
1
, M
n−1
1,1
+
N
n=1
I
X
1,n
; Z
1,n
|W
1
, W
2
, M
n−1
1,1
, Y
n−1
1,1
(39)
剩余24页未读，继续阅读
weixin_38750007
- 粉丝: 4
- 资源: 921
我的内容管理
展开
最新资源
- 新型矿用本安直流稳压电源设计：双重保护电路
- 煤矿掘进工作面安全因素研究：结构方程模型
- 利用同位素位移探测原子内部新型力
- 钻锚机钻臂动力学仿真分析与优化
- 钻孔成像技术在巷道松动圈检测与支护设计中的应用
- 极化与非极化ep碰撞中J/ψ的Sivers与cos2φ效应：理论分析与COMPASS验证
- 新疆矿区1200m深孔钻探关键技术与实践
- 建筑行业事故预防：综合动态事故致因理论的应用
- 北斗卫星监测系统在电网塔形实时监控中的应用
- 煤层气羽状水平井数值模拟：交替隐式算法的应用
- 开放字符串T对偶与双空间坐标变换
- 煤矿瓦斯抽采半径测定新方法——瓦斯储量法
- 大倾角大采高工作面设备稳定与安全控制关键技术
- 超标违规背景下的热波动影响分析
- 中国煤矿选煤设计进展与挑战：历史、现状与未来发展
- 反演技术与RBF神经网络在移动机器人控制中的应用
资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！
点击此处反馈
