Uplink-downlink duality for integer-forcing

Consider a MIMO uplink channel with channel matrix H and a MIMO downlink channel with channel matrix Hτ. 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 Hτ to L single-antenna receivers with target integer matrix Aτ. 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.