A Further Consideration on the HK and the CMG regions for the Interference Channel

We present a simplified form of the HK (= HanKobayashi) achievable rate region for the general interference channel by using Fourier-Motzkin algorithm, as well as a simplified form of the HK region without the don’t care inequalities. Also, in the same spirit, we give an improved form of the CMG (= Chong-Motani-Garg) region. I. THE ORIGINAL HK ACHIEVABLE REGION Let ω(y1y2|x1x2) be an interference channel with two inputs x1, x2 and two outputs y1, y2. In order to establish an achievable rate region for the general interference channel, Han and Kobayashi [1] started with the introduction of a preparatory rate region SHK(Z) (for each Z ∈ P∗) of (S1, T1, S2, T2)’s that are expressed in terms of four auxiliary random variables U1, U2,W1,W2 and a time sharing parameter Q. Here, P∗ is the set of all Z = QU1W1U2W2X1X2Y1Y2 such that i) U1, W1, U2,W2 are conditionally independent given Q; ii) X1 = f1(U1,W1|Q), X2 = f2(U2,W2|Q) ; iii) Pr{Y1 = y1, Y2 = y2|X1 = x1, X2 = x2} = ω(y1y2|x1x2), where f1(·, ·|·) and f2(·, ·|·) are deterministic functions. With these notations, SHK(Z) is defined as the set of all (S1, T1, S2, T2) such that S1 ≤ a1, (1) T1 ≤ b1, (2) T2 ≤ c1, (3) S1 + T1 ≤ d1, (4) S1 + T2 ≤ e1, (5) T1 + T2 ≤ f1, (6) S1 + T1 + T2 ≤ g1, (7) −S1 ≤ 0, (8) −T1 ≤ 0, (9)