Channel diversity needed for vector interference alignment

In this paper, 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. The channel diversity L is modeled by L independently fading real-valued parallel channels. Existing results in the literature for K = 3 show that the optimal 1/2 degrees of freedom per user can be approached at the speed of 1/L (i.e. the gap to 1/2 degrees of freedom per user decreases inversely proportional to L). In this paper, we show that when K ≥ 4, the speed of convergence is significantly slower. In particular, the gap to 1/2 degrees of freedom per user can decrease at most like 1/√L. Furthermore, when K is of the order of √logL, the speed of convergence is smaller than 1√(L)⁴.