On Capacity of Non-Coherent Diamond NetworksThis work was supported in part by NSF grants 1514531 and 1314937

There is a vast body of work on the capacity bounds for a "coherent" wireless network, where the network channel gains are known, at least at the destination. However, there has been much less attention to the case where the network parameters (channel gains) are unknown to everyone, i.e., the non-coherent wireless network capacity. In this paper, we study the generalized degrees of freedom (gDoF) of the block-fading non-coherent diamond (parallel relay) network with asymmetric distributions of link strengths, and a coherence time of T symbol duration. We first derive an outer bound for this channel and then derive the optimal signaling structure for this outer bound. Using the optimal signaling structure we solve the outer bound optimization problem for gDoF. Using insights from our outer bound signaling solution, we devise an achievability strategy based on a novel scheme that we call train-scale quantize-map-forward. This uses training in the links from source to relays, scaling and quantizing at relays combined with non-training based schemes. We show the optimality of this scheme with respect to the outer bound in terms of gDof. In non-coherent point-to-point MIMO, where the fading channel is unknown to transmitter and receiver, an important trade-off between communication and channel learning was revealed by Zheng and Tse, by demonstrating that not all antennas available might be used as it is sub-optimal to learn all their channel parameters. Our results in this paper for the diamond network demonstrates that in certain regimes the optimal scheme uses a sub-network, demonstrating a similar trade-off between channel learning and communications. However, in other regimes it is useful to use the entire network and not use training at all in the signaling, as traditional training based schemes are sub-optimal in these regimes.