Capacity theorems for the relay channel

A relay channel consists of an input x,, a relay output yl, a cJmnnel output y, and a relay sender x2 (whose trasmission is allowed to depend on the past symbols y,). l%e dependence of the received symbols upm the inpnts is given by p(y,y,lx,,x,). ‘l%e channel is assumed to be memoryless. In this paper the following capacity theorems are proved. 1) Ifyisadegnukdformofy,,the.n C-m=Phx2) min(l(X,,X,; Y),I(X,; Y,lX&). 2) Ify,isadegradedformofy,tben C==maxpcx,) m=JXI; Ylx2). 3) If p(y,yllx,,x2) is an arWnuy relay chaonel 4th feedback from Crud to both x1 and ~2, then C=maXpcx,.x*, min(W,,X*; Y)J(x,; y9 Y,lX3). 4) For a general relay channel, C< max+,,,d min(Z(Xl,X2; YMX,; K Y,lXJ. Superposition block Markov encoding is used to show achievabiiity of C, and converses are established. ‘Ihe capacities of the Gaussian relay dumWI and certain discrete relay channels are evaluated. Finally, an acbievable lower bound to the capacity of the general relay channel is established. Manuscript received December 1, 1977; revised September 28, 1978. This work was partially supported by the National Science Foundation under Grant ENG76-03684, JSEP NooOl6-67-A-oOI2-oSdI, and Stanford Research Institute Contract DAHC-15-C-0187. This work was presented at the International Symposium on Information Theory, Grigano, Italy, June 25-27, 1979. T. M. Cover is with the Departments of Electrical Engineering and Statistics, Stanford University, Stanford, CA 94305. A. A. El Gamal is with the Department of Electrical Engineering Systems, University of Southern California, University Park, Los Angeles, CA 90007.