General broadcast channels with degraded message sets

A broadcast channel with one sender and two receivers is considered. Three independent messages are to be transmitted over this channel: one common message which is meant for both receivers, and one private message for each qf them. The coding theorem and strong converse for this communication situation is proved for the case when one of the private messages has rate zero. I. I -NTRODUCTION W E CONSIDER a two-receiver broadcast channel defined by T. M . Cover [l] as a pair of discrete memoryless channels (V, W) with common input alphabet Y and respective output alphabets X and 2. (We use the same symbol for discrete memoryless channels and for their transition probability matrices, and we suppose that all alphabets are finite.) The nth memoryless extension of this broadcast channel is defined by the pair (VI”, W ”), where, e.g., foryn = YIYZ “‘Yn E Y”,x~=x1x2”‘xn E X”. An (n,t)-code for this channel is given by codewords yjnkl E Y n (1 _< j 5 M1, 1 _< k 5 M2, 1 5 1 < MO); and corresponding decoding sets 3Qjl c X”,,@kl c 2” such that both (&jl{ and (@hll are disjoint families, and Vn(AjllYjnkl) 2 1 t, w”(@)klIy~~~) L 1 e for all j,/z,l. A triple of nonnegative numbers (R~,Rz,Ro) is called an E-achievable rate triple for this channel, if, for any 6 > 0 and large enough n, there exists an (n,t)-code (yj”kl, &jl, @kl; 1 < j < MI, 1 5 K 5 Mz, 1 5 1 < MO] satisfying n-l.logMi 1 Ri 6, i = 1,2,0. The determination of all the t-achievable rate triples for a general two-receiver broadcast channel is still an open problem. A number of partial results are available, including the complete solution of the so-called “degraded” case. (See [2]-[8]. References [5] and [8] contain essentially the same results, however, only [5] contains complete proofs. Therefore, we shall refer to [5] in the sequel.) In this paper we describe all the pairs (Rl,Ro) such that (Rl,O,Ro) is an t-achievable rate triple. We denote by B(E) the set of all such pairs, and let .%! P ne>o n(6). Notice that, if channel W is a degraded version of V, then a rate triple (Rl,Rz,Ro) is c-achievable (for the general broadcast problem) if and only if (Rl,Ro + R2) E 33(c). Manuscript received December 29,1975; revised April 29,1976. The authors are with the Mathematical Institute of the Hungarian Academy of Sciences, 1053 Budapest, ReBltanoda-u.l3-15, Hungary. (See also [7].) We now formulate the problem and state the main result. Definition: (Code for the general broadcast channel with degraded messages.) An (n,c)-code for the general discrete memoryless broadcast channel with degraded messages is given by codewords y,“l E Y n (1 I j < Ml, 1 I I I MO) and corresponding decoding sets SQjl c X”, @ l c 2” such that both (3Qjl) and (el] are disjoint families, and Vn(Ajllyjnl) 2 1 E, Wm(e!,lyjnl) 1 1 6, (1) for all j,Z. A pair of nonnegative numbers (Rl,Ro) is called an Eachievable rate pair for the broadcast channel with degraded messages if, for any 6 > 0 and large enough n, there exists an (n,e)-code ((y$3qjl,@l); 1 5 j 5 Ml, 1 5 1 5 MO) satisfying n-1 log Mi 2 Ri 6, i = l,O. Denote by R(t) the set of all the t-achievable rate pairs. (It is easily seen that this coincides with the previous definition of B(t).) A pair of nonnegative numbers (Rl,Ro) is called an achievable rate pair if it is t-achievable for every t > 0. We write Yl = n Yi?R(t). c>o For a quadruple of random variables (r.v.‘s) (U, Y,X,Z), we write (U,Y,X,Z) E P if U,Y and the pair (X,2) form a Markov chain in this order, and if the conditional distributions of X given Y and of Z given Y are defined by V and W, respectively. Here and hereafter, all r.v.‘s have finite range. For an r.v. U, (1 UII denotes the cardinality of the range of U. Theorem: Let 9 A {(R1,Ro); RI I I(Y A XI U), R. I I(U A Z), RI + Ro 5 ICY A X), (U,Y,X,Z) E PD, IiUiI 5 I/Y/I + 4; then, for every t > 0, 92(t) = n = 9. Remarks: 1) The interesting part of this theorem is the converse, (i.e., that no rate pairs outside 9 are achievable for any t > 0). This will be deduced from [9, theorem 21. The direct part is trivial for those familiar with Bergmans’ proof of the coding theorem for degraded broadcast channels [4]. 2) We shall show in the Appendix that the rates found by Cover [5] and van der Meulen [B] for this problem exhaust 3. Ki iRNER AND MARTON: GENERAL BROADCAST CHANNELS 61