Uplink-downlink duality for integer-forcing

Consider a MIMO uplink channel with channel matrix H and a MIMO downlink channel with channel matrix H^T. It is well-known that any rate tuple that is achievable on the uplink is also achievable on the downlink under the same total power constraint, i.e., there is an uplink-downlink duality relationship. In this paper, we consider the integer-forcing strategy, in which users steer the channel towards an integer-valued effective channel matrix so that the receiver(s) can decode integer-linear combinations of the transmitted codewords. Recent efforts have demonstrated the benefits of this strategy for uplink, downlink, and interference alignment scenarios. Here, we establish that uplink-downlink duality holds for integer-forcing. Specifically, in the uplink, L transmitters communicate over channel matrix H to an L-antenna receiver with target integer matrix A. In the downlink, an L-antenna transmitter communicates over channel matrix H^T to L single-antenna receivers with target integer matrix A^T. We show that any computation rate tuple that is achievable in the uplink is achievable for the same total power in the downlink and vice versa.