The asymmetric DoF Region for the 3-user MxN Interference Channel

In this paper, the 3-user Gaussian MIMO interference channel is considered. The asymmetric distribution of the DoF, where different users have different number of DoF, is studied. Two cases are presented, the first is when all transmitters and receivers have equal number of antennas M , the other when the transmitters have M antennas each, while the receivers have N antennas each. It is assumed that the channel coefficients are constant and known to all transmitters and receivers. The region of the achievable DoF tuple (d1; d2; d3) is presented. I. SUMMARY The DoF region of 3-user M × N Gaussian MIMO interference channel was presented in [1],[2], where [2] derived the upperbound and showed that the scheme presented in [1] is DoF optimal. All previous work assumed that all transmitters have the same number of DoF. The need to identify the asymmetric DoF region comes from the reality the different nodes require different rates and have different channel strength at different times. We consider the 3 user M ×N interference channel as defined in the system model of [1]. We assume full CSI knowledge at all transmitters. We use the notation < A >b to for the modulo of A to the base b. First we consider the case when M = N . Let di be the DoF of user i and Ii be the interference at the ith receiver, we assume without loss of generality that First we consider the case when M = N . Let di be the DoF of user i and Ii be the interference at the ith receiver, we assume without loss of generality that d1 ≥ d2 ≥ d3 (1) For user 1 to be able decode its message, the signal space and interference space must be distinct at receiver 1 or, d1 + I1 ≤M (2) I1 = max(d2, d3) (3) d1 + max(d2 + d3) ≤M (4) The interference space size is dominated by d1 at receivers two and three because d1d1 ≥ max(d2, d3) then, I2 = I3 = d1 (5) (6) Then the DoF region of the asymmetric case is, di + dj ≤M, i = 1, 2, 3, j = 1, 2, 3, i 6= j (7) From (2,7), it is easy to see that an increase of d to di would result in a decrease in the DoF of other users, each by d or a total loss in the sum DoF by d. Next. we consider the general case M 6= N . We assume-without loss of generality-that M is larger than N. In the proposed schemes, the nullspaces of the channels to the non intended receivers are used in combination with interference alignment to This work was supported in part by a grant from Qatar national research fund . Mohammed Nafie is also affiliated with Faculty of Engineering, Cairo University. ar X iv :1 40 7. 29 81 v1 [ cs .I T ] 1 0 Ju l 2 01 4