An achievable rate region for the broadcast channel

Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(y_1, y_2 x)</tex> denote a discrete memoryless channel with a single source <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> and two independent receivers <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y_1</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y_2</tex> . We exhibit an achievable region of rates <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(R₁₁,R₁₂,R₂₂)</tex> at which independent information can be sent, respectively, to receiver 1, to both receivers 1 and 2, and to receiver 2. The achievability of the region is shown by using a version of the asymptotic equipartition property involving many simultaneous "typicality" constraints. These results immediately generalize to yield an achievable rate region for the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> -sender <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -receiver channel in terms of standard mutual information quantities.