On the degrees of freedom achievable through interference alignment in a MIMO interference channel

Consider a K-user flat fading MIMO interference channel where the k-th transmitter (or receiver) is equipped with M_k (respectively N _k) antennas. If a large number of statistically independent channel extensions are allowed either across time or frequency, the recent work [1] suggests that the total achievable degrees of freedom (DoF) can be maximized via interference alignment, resulting in a total DoF that grows linearly with K even if M_k and N_k are bounded. In this work we consider the case where no channel extension is allowed, and establish a general condition that must be satisfied by any degrees of freedom tuple achievable through linear interference alignment. When M_k = M and N_k = N for all k, this condition implies that the total achievable DoF cannot grow linearly with K, and is in fact no more than M + N - 1. If, in addition, all users have the same DoF d = 1, then this upper bound on the total DoF is actually tight for almost all MIMO interference channels.