Channel Diversity needed for Vector Space Interference Alignment

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 non-degenerate 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 $K\geq4$, the gap is significantly larger. In particular, the gap to the optimal $1/2$ degrees of freedom per user can decrease at most like $1/\sqrt{L}$, and when $L$ is smaller than the order of $2^{(K-2)(K-3)}$, it decays at most like $1/\sqrt[4]{L}$.