648 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-26, NO. 6, NOVEMBER 1980 Multiple Access Channels with Arbitrarily Correlated Sources THOMAS M. COVER, FELLOW, IEEE, ABBAS EL GAMAL, MEMBER, IEEE, ANDMASOUD SALEHI, MEMBER,IEE Absrmcr-Let {(q, r$)}#L* be a source of independealt identicauy distributed (i.i.d.) disc&e random variables with joint probability mass function p(u,o) and common part w-f(u)=g(u) in the sense of Witsenbawn, Gacs, and Kkner. It is shown that such a source can be sent with arbitrarily small probability of error over a multiple access ChaMel (MAC) {Xl X~*9%P(Yl~,, X,)>> with allowed codes {q(u), x2(w)] if there exist probability mass functions P(S),P(X,lS, U),P(X,lS9 u)9 s’dl that H(UIV)<Z(X,;YIX,,~,S), H(VIU)<Z(X*;YIX,,U,S), H(U,VIW)<Z(X,,X,;YIW,S), H(u,~)<z(x,,x,;n P(s,u,u,x,,~z,Y)=P~~~P~~,~~P~~,l~,~~P~~*l~~~~P~Yl~,~~*~. ‘zbls region inch&s the multiple aaxw channel region and the Slepian- Wolf data compression region as special cases. I. INTRODUCTION T HE MULTIPLE access channel (MAC) p(u Ix,, x2) has a capacity region [l], [2] given by the convex hull of all (R,, R,) satisfying, for somep(x,, x,)=p(x,)p(x,), the inequalities R, (1(X,; Ylx,), R,WX,; YIX,), R,+R,<z(X,,X,;Y). (1) Supposenow that the source U for X, and V for X, are correlated according to p(u, u). It follows easily that U Manuscript received November 28, 1978; revised February 28, 1980. This work was supported in part by the National Science Foundation under Grant ENG 76-23334, in part by the Stanford Research Institute under International Contract D/&C-15-C-0187, and in part by the Joint Scientific Enaineerina Program under Contracts NO001475-C-0601 and F44620-76-C&01. This paper was presented at the 1979 IEEE Intema- tional Symposium on Information Theory, Grignano, Italy, June 25-29, 1979. T. M. Cover is with the Departments of Electrical Engineering and Statistics, Stanford University, Durand Building, Room 121, Stanford, CA 94305. A. El Gamal was with the Department of Electrical Engineering, University of Southern California, University Park, Los Angeles, CA. He is now with the Department of Electrical Engineering, Stanford Univer- sity, Stanford, CA 94305. M. Salehi was with the Department of Electrical Engineering, Stan- ford University, Stanford, CA. He is now with the Department of Electrical Engineering, University of Isfahan, Isfahan, Iran. and I’ can be sent over the multiple access channel if, for someAxI, ~~)=Ax&(x2)~ H(U)<Z(X,; YIX,), fw)<Z(X,; YIXA H(U)+H(v)<z(X,, x2; Y). (2) In this paper, we increase this achievable region in two ways: 1) the left side will be made smaller’, and 2) the right side will be made larger by allowing X, and X, to depend on U and V and thereby increasing the set of mass distributions p(x,, x2). It will be shown (see Theorem 1 for a precise and more general statement) that U and V can be sent with arbitrarily small error to Y if fwIV)ax,; YlX2,O H(VIU)<Z(X,; YIX,,U), wu, V)<W,, x2; Y), (3) for some p(u, 0, xi, x2, u) =A% ~MX,lU)P(X,l~) .p(ylx,, x2). This result can be further generalized to sources (U, V) with a common part W=j( U) = g( V). The following theorem is proved. Theorem 1: A source (V, V)NII~P(U~,U~) can be sent with arbitrarily small probability of error over a multiple access channel {%i xX2, 3, p(yIx,, x2)}, with allowed codes {x,(u), x2(u)} if there exist probability mass func- tionsp(s), p(x,]s, u), p(x,ls, u), such that H(UJV)<Z(X,; YJX;?,V, S), H(VIU)<I(X,; YIX,,U, S), fqUJqW)<W,, x2; YIW, a, H(U,V)<Z(X,, x2; Y), (4) where p(s, u, u, xi, x2, Y> =P(s>P(u, U)P(X,lUT s) *P(~,I~JlP(Yl~l~ 3). Remark I: The region described above is convex. Therefore no time sharing is necessary.The proof of the convexity is given in Appendix B. Remark 2: It can be shown that if error-free transmis- sion is possible, then in order to generate a random code for error-free transmission, it is enough to consider those auxiliary random variables S whose cardinality is bounded above by ~~~ll~~ll~II~211~II~II~. ‘This improvement could be obtained from the results of Slepian and wolf 131. 0018-9448/80/1100-0648$00.75 0 1980 IEEE