1 On Gaussian Interference Channels with mixed interference Yang Weng and Daniela Tuninetti University of Illinois at Chicago, yweng3@uic.edu and danielat@uic.edu Abstract— This work analyzes a particular achievable region for Gaussian interference channels (IFC) derived from the gen- eral Han-Kobayashi region. By reformulating the Han-Kobayashi achievable region as the sum of two sets, we characterize the maximum achievable sum-rate with Gaussian inputs and without timesharing in closed from for any Gaussian IFC. We then show that the computed sum-rate meets the upper bound by Kramer for any IFC with mixed interference, and not only for IFC with strong interference. We then show that for a certain subclass of IFCs with mixed interference, the capacity region contains a line segment of slope -1 of which we characterize the extreme points in term of the power allocation among private and common messages. Finally, for another subclass of IFCs with mixed interference, we show that the capacity region coincides with that of the corresponding Z-IFC, if Gaussian inputs are used. I. I NTRODUCTION A two-user IFC consists of two input alphabets (X 1 , X 2 ), two output alphabets (Y 1 , Y 2 ), and a channel transition prob- ability P Y1 Y2|X1 X2 . The channel is assumed memoryless. Transmitter u, u ∈{1, 2}, has a length-n message W u , uni- formly distributed over the set {1, ··· , e nRu }, for receiver u, where n denotes the codeword length and R u the transmission rate in nats per channel use. Decoder u, u ∈{1, 2}, outputs the estimate  W u (Y n u ) of its intended message W u . The capacity region of the IFC is the closure of the set of rate pairs (R 1 ,R 2 ), such that the receivers can decode their intended message with arbitrarily small error probability as n →∞. A two-user Gaussian IFC in standard form is defined as Y 1 = X 1 + √ aX 2 + Z 1 , a ∈ R + Y 2 = √ bX 1 + X 2 + Z 2 , b ∈ R + Z u ∼N (0, 1), E[|X u | 2 ] ≤ P u , u = {1, 2}. Following [2], we say that the Gaussian IFC has strong interference if a ≥ 1 and b ≥ 1, mixed interference if a ≥ 1 and b< 1, or a< 1 and b ≥ 1, and weak interference if a< 1 and b< 1. If a =0 or b =0, the channel is referred to a Z-IFC. If ab =1, the channel is referred to a degraded IFC. The capacity region of Gaussian IFC is known strong interference (for which it is optimal for each user to decode both messages), and for Z-IFC when the non-zero channel coefficients is larger or equal to 1. For a general IFC, the largest known achievable region is due to Han and Kobayashi [4], HK for short in the follow- ing. The ingredients of the HK scheme are: rate-splitting, superposition coding and jointly decoding. Each user spits its message into two parts W u =(W u0 ,W uu ), u ∈{1, 2}, where W u0 –the “common” message– is to be decoded at both receivers, while W uu –the “private” message– is to be decoded at intended receiver only. At the encoder side, the common and the private messages are superimposed. At the decoder side, the two common messages and the intended private message are jointly decoded. In [3] it is shown that the probabilities of error for the HK scheme can be driven to zero if the rates R u = R u0 + R uu , u ∈{1, 2}, satisfy R 11 ≤ I (X 1 ; Y 1 |U 1 ,U 2 ,Q) (1a) R 11 + R 20 ≤ I (X 1 ,U 2 ; Y 1 |U 1 ,Q) (1b) R 11 + R 10 ≤ I (X 1 ; Y 1 |U 2 ,Q) (1c) R 11 + R 10 + R 20 ≤ I (X 1 ,U 2 ; Y 1 |Q) (1d) R 22 ≤ I (X 2 ; Y 2 |U 1 ,U 2 ,Q) (1e) R 22 + R 10 ≤ I (X 2 ,U 1 ; Y 2 |U 2 ,Q) (1f) R 22 + R 20 ≤ I (X 2 ; Y 2 |U 1 ,Q) (1g) R 22 + R 20 + R 10 ≤ I (X 2 ,U 1 ; Y 2 |Q) (1h) for all distributions of the form P QU1X1U2X2 = P Q P U1X1|Q P U2X2|Q . (2) In [3], the region (1) is compactly expressed in terms of rate- bounds for R 1 , R 2 , R 1 + R 2 , 2R 1 + R 2 and R 1 +2R 2 only by using Fourier-Motzkin elimination, and the cardinality of the time-sharing random variable Q is shown to be at most 8. An exhaustive evaluation of the HK region in (1) over all possible input distributions in (2) is prohibitively complex. For Gaussian IFC, subregions of the HK region amenable to evaluation have been considered in [6]–[8]. In [8], it is shown that with Gaussian inputs, the cardinality of Q can be limited to 5. The best know outer bound to the capacity of Gaussian IFC are due to Kramer [5], Etkin et al. [2], and Shang et at [9]. The results in [5, Th.1], [2], and [9] are based on giving the receivers some side information such that the capacity of the enhanced channel can be computed. The best known upper bound on the sum-rate can be obtained by combining the results of [5, Th.1], [2], and [9]. The result in [5, Th.2] is valid when at least one of the channel parameters is smaller than one; it based on the idea of transforming the original channel into a degraded IFC, then let the transmitter cooperate so as to form a degraded broadcast channel. A combination of [5, Th.1] and [5, Th.2] gives the best outer bound for IFC with mixed interference.