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 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.