Multiple access channels with arbitrarily correlated sources

Abstract

Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{(U_{i},V_{i})\}_{i=1}^{n}</tex> be a source of independent identically distributed (i.i.d.) discrete random variables with joint probability mass function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(u,v)</tex> and common part <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w=f(u)=g(v)</tex> in the sense of Witsenhausen, Gacs, and Körner. It is shown that such a source can be sent with arbitrarily small probability of error over a multiple access channel (MAC) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{\cal X_{1} \times \cal X_{2},\cal Y,p(y|x_{1},x_{2})\},</tex> with allowed codes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{x_{l}(u), x_{2}(v)\}</tex> if there exist probability mass functions <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(s), p(x_{1}|s,u),p(x_{2}|s,v)</tex> , such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(U|V)&lt;I(X_{1}; Y|X_{2},V,S),</tex> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(V|U )&lt;I(X_{2};Y|X_{1},U,S),</tex> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(U,V|W)&lt;I(X_{1},X_{2};Y|W,S),</tex> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(U,V)&lt;I(X_{1},X_{2};Y),</tex> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\mbox{where}</tex> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(s,u,v,x_{1},x_{2},y), Xl, X2, y)=p(s)p(u,v)p(x_{1}|u,s)p(x_{2}|v,s)p(y|x_{1},x_{2}).</tex> lifts region includes the multiple access channel region and the Slepian-Wolf data compression region as special cases.