Degraded Broadcast Channels

Abstract

Let A = {1,..., a}, B = {1,..., b}, and C = {1,..., c}. Let w₁(· | ·) be the c.p.f. of a d.m.c. A → B, and w₂(· | ·) the c.p.f. of a d.m.c. B → C. Write A ^*_n for the Cartesian product of n A’s, with similar definitions for B ^*_n , C ^*_n , and, later, for D ^* _n . There are three individuals I, II, III who act as follows: I wishes to send to II an arbitrarily chosen pair (i, j) of messages over the d.m.c. w₁(·|·), and the message j to III over the d.m.c. w₃(·|·), where

$$ {\omega _3}\left( {{s_2}\left| {{s_1}} \right.} \right) = \sum\limits_{s = 1}^b {{\omega _2}} \left( {{s_2}\left| s \right.} \right){\omega _1}\left( {s\left| {{s_1}} \right.} \right) $$

$$ 1\underline{\underline < } \,{s_1}\underline{\underline { < \,}} \,a,1\underline{\underline < } \,{s_2}\underline{\underline < } \,c $$

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