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 kth transmitter (or receiver) is equipped with M-k (respectively N-k) antennas. If an exponential (in K) number of generic channel extensions are used either across time or frequency, Cadambe and Jafar [1] showed 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 (d(1), d(2), ... d(K)) achievable through linear interference alignment. For a symmetric system with M-k = M, N-k = N, d(k) = d for all k, this condition implies that the total achievable DoF cannot grow linearly with K, and is in fact no more than K (M + N)/(K + 1). We also show that this bound is tight when the number of antennas at each transceiver is divisible by d, the number of data streams per user.