Degrees of freedom achieved using subspace alignment chains for three-cell networks

In this paper we extend the notion of subspace alignment chains (SACs) for the three-user multiple-input multiple-output (MIMO) interference channel to three-cell MIMO cellular networks. By extending the notion of SACs to three-cell networks we show that when d ϵ Z
⁺
DoF/user are achievable in a three-user M × N interference channel (IC) using linear beamforming, then any DoF-tuple {d
_ij
}, where d
_ij
ϵ Z
⁺
represents the DoF of the jth user in the ith cell, that satisfies Σ
_j=1
^k
d
_ij
≤ d ∀ i is achievable in a three-cell MIMO cellular network with K users per cell having M antennas per user and N antennas per base station (i.e., the (3, K, M, N) network) using linear beamforming. When restricted to symmetric DoF, this result states that whenever d DoF/user are achievable in a three-user M × N IC with d = rs, for some r, s ϵ Z
⁺
, then r DoF/user are achievable in the (3, s, M, N) network. Although the DoF achieved using SACs is not necessarily the largest possible, they are established through a constructive procedure where we show how SACs designed for the three-user interference channel can be modified to design transmit beamformers for the three-cell MIMO cellular networks. Further, we highlight the role played by redundant antennas in reducing the computational cost of designing transmit beamformers for interference alignment.