The capacity of the semideterministic relay channel

The capacity of the class of relay channels with sender xl, a relay sender x2, a relay receiver y, = /(x1, x2), and ultimate receiver y is proved to be The relay channel (%, X ‘&, p(y,y, 1 x,, x,), 9 X 9,) is a model for the communication between a sender x, and a receiver y through two paths; a direct path from x, to y, and a path from Manuscript received September 5, 1979; revised June 16, 1981. This work was supported in part by the Joint Services Electronics Program through the Air Force Office of the Scientific Research (AFSC) under Contract F44620-76C-0061, and in part by the National Science Foundation under Grant ENG 78-23334. This paper was partially presented at the International Symposium on Information Theory, Grignano, Italy, 1979. Abbas El Gamal is with Stanford University, Stanford, CA 94305. Mohammad Aref was with Stanford University, Stanford, CA 94305. IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-28, NO. 3, MAY 1982 x, to y through the relay (y,, x2). .The relationship among the received symbols y and y, and the transmitted ones x, and x2 is given by the probability transition matrix p( y, y, 1 x,, x2). The problem of finding the capacity of the relay channel (i.e., maximum rate of transmission from x, to y) was first studied by van der Muelen [l]. In [2] Cover and El Gamal established the capacity when the relay channel is degraded, reversely degraded, and when feedback is added from the receivers y and y, to both senders x, and x2. A general upper bound to capacity and an achievable rate (lower bound) were also established in [2]. In this note, we show that a special case of the lower bound to the capacity given in [2, theorem 71 is in fact the capacity when the relay receiver y, is a deterministic function of x, and x2. First, recall the following version of [2, theorem 71. Theorem 7 [2]: For any relay channel, the following rate R* is achievable: R*=sup{min{l(U;Y,IX2,V)+1(X,;Y,Y,IX2,U),I(V;Y) +z(u; YI x,, q + z(x,; y, 6 I x2, u)}}, (1) where the supremum is taken over all joint probability mass functions of the form P~~,~~~I~~2~Y~Y,~~,~=P~~~~~I~~P~~,I~~P~~2I~~ .P(Y, Y, I x,9 X,)P(3, I x2, Y,, u>, (4 subject to the constraint I(?,; Y, I Y, x&J;) SZ(X,; YI v). (3) Now, substituting Y, = 0, Y = X,, and U = (X2, Q) in (I)(3), we obtain the following theorem. Theorem: For any relay channel, the following rate R, is achievable: R. = SUP min{Z(X,, X2; Y>, Z + Z(x,; YI X2, Q)}, (4) where the supremum is over all probability mass functions of the form PC49 %x2+Y,) =P(4,x,,x2)P(Y,Y,l~,,~2), (5) and Q is an arbitrary random variable. Corollary: If y, is a deterministic function of x, and x2, then C=R,=~~~~~{Z(X,,X,;Y),H(Y,IX,) +Z(X,;YlX2,Y,)) (6) Proof: That (6) is a special case of R, can be easily seen by setting Q = Y,. The converse part of the corollary follows easily from the outer bound of [2, theorem 41, and the fact that if Y, =f(x,, x2>, then 1(X,; Y, Y, I X2) = WY, I X2) + 1(X,; YI X2, Y,), and the corollary is proved. Remarks: If both y and y, are deterministic functions of (x,, x2), i.e., if the relay channel is deterministic, then it follows from the corollary that the capacity is given by c=~~~:{H(Y),H(Y,Y,IX,)}. ACKNOWLEDGMENT The author is indebted to the referees for their invaluable remarks during the preparation of this paper.