C hannel Diversity Needed for Vector Space Interference Alignment
Abstract: We consider vector space interference alignment strategies over the K-user interference channel and derive an upper bound on the achievable degrees of freedom as a function of the channel diversity L, where the channel diversity is modeled by L real-valued parallel channels with coefficients drawn from a nondegenerate joint distribution. The seminal work of Cadambe and Jafar shows that when L is unbounded, vector space interference alignment can achieve 1/2 degrees of freedom per user independent of the number of users K. However, wireless channels have limited diversity, in practice, dictated by their coherence time and bandwidth, and an important question is the number of degrees of freedom achievable at finite L. When K = 3 and if L is finite, Bresler et al. show that the number of degrees of freedom achievable with vector space interference alignment is bounded away from 1/2, and the gap decreases inversely proportional to L. In this paper, we show that when K ≼ 4, the gap is significantly larger. In particular, the gap to the optimal 1/2 degrees of freedom per user can decrease at most like 1/√L, and when L is smaller than the order of 2(K-2)(K-3), it decays at most like 1/4√L.